Digital Filters. Linearity and Time Invariance. Linear Time-Invariant (LTI) Filters: CMPT 889: Lecture 5 Filters

Size: px
Start display at page:

Download "Digital Filters. Linearity and Time Invariance. Linear Time-Invariant (LTI) Filters: CMPT 889: Lecture 5 Filters"

Transcription

1 Digital Filters CMPT 889: Lecture 5 Filters Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University October 7, 29 Any medium through which a signal passes may be regarded as a filter. Typically however, a filter is viewed as something which modifies the signal in some way. Examples include: stereo speakers our vocal tract our musical instruments A digital filter is a formula for going from one digital signal to another. x(n) Figure : A black box filter. y(n) CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 2 Linearity and Time Invariance Linear Time-Invariant (LTI) Filters: A filter is linear if it satisfies the following two properties:. Scaling: the amplitude of the output is proportional to the amplitude of the input (i.e., scaling can be done either at the input or the output) L{gx( )} = gl{x( )} 2. Superposition: the output due to a sum of input signals is equal to the sum of outputs due to each signal alone. L{x ( ) + x 2 ( )} = L{x ( )} + L{x 2 ( )} A filter is time-invariant if its behaviour is not dependent on time: if the input signal is delayed by N samples, the output waveform is simply delayed by N samples: L{x( N)} = L n N {x( )} = y(n N) Liner Time-Invariant (LTI) filters modify the amplitude and/or phase of the spectral components of an input signal. LTI filters are guaranteed to produce a sinusoid in response to a sinusoid, without altering the frequency. They may only attenuate, reject, amplify (boost) or leave unchanged, any existing frequency components of the input signal. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 3 CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 4

2 Filter Frequency Response Different Filter Types The characteristic of a filter is described by its frequency response which consists of both a magnitude and a phase response:. magnitude response Describes the gain of a filter at every frequency: a positive gain boosts the signal while a negative gain attenuates the signal. Determined by the ratio of the peak output amplitude to the peak input amplitude at a given frequency. 2. phase response: Determined by subtracting the input phase from the output phase at a particular frequency. A frequency response describes how the amplitude and phase of a sound s spectral components are changed by the filter. The region where frequencies are permitted to pass is called the pass band and the region where frequencies are attenuated is called the stop band. There are four (4) basic types of filters:. Low-pass: permits frequencies below a cutoff frequency f c to pass with little change while attenuating or rejecting frequencies above f c. 2. High-pass: permits frequencies above f c to pass while attenuating those frequencies which fall below. 3. Band-pass: passes frequencies within a certain frequency band. 4. Band-reject: attenuates or rejects frequencies within a certain frequency band (also called a notch filter). CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 5 CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 6 Magnitude Response of Basic Filters Quality Factor No digital filter is ideal and therefore they will always a have smooth (rather than a clearly defined or discontinuous) transition between the pass and stop bands at the cutoff frequency f c. magnitude (db) Low Pass 3 3 fc frequency magnitude (db) High Pass fc frequency A pass-band filter is characterized by its quality factor Q which is inversely proportional to its bandwidth Q = f c BW where f c is the center frequency of the band. A high quality factor (a high Q) denotes a filter with a narrow bandwidth and a low Q denotes a filter with a wide bandwidth. magnitude (db) 3 Band pass fc BW frequency magnitude (db) 3 Band reject (notch) fc BW frequency Figure 2: Amplitude response of different filter types. The cutoff frequency is typically defined as the frequency at which the power transmitted by the filter drops to one-half (by -3 db) of the maximum power transmitted in the passband. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 7 CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 8

3 Delay Review The Ideal Low-Pass Filter Recall that whenever a signal can be expressed in the form x(t) = s(t t ), we say that x(t) is a time-shifted version of s(t). If t is a positive number, then the shift is to the right and we say that the signal has been delayed in time. If t is a negative number, x(t) effectively becomes x(t) = s(t + t ), the shift is to the left and Recall that a lowpass filter is one that allows low frequencies to pass, while attenuating anything above a specified cutoff frequency f c. The amplitude response of the ideal low-pass filter is given in the figure below: Gain Frequency Figure 3: Amplitude Response for an ideal low-pass filter. fc fs/2 the signal has been advanced in time. Unfortunately such an ideal amplitude response is not realizable in discrete time. How close we come depends on the complexity of the filter. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 9 CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 A Simple Low-Pass Filter Intuitive Analyis at Low Frequencies The simplest low-pass filter (and therefore furthest from ideal) is given by the difference equation y(n) = x(n) + x(n ). The corresponding system diagram is given below: x(n) z y(n) Figure 4: System diagram for the simple low-pass filter y(n) = x(n) + x(n ). This filter is really just a running averager (2-point averager) with a factor of two. That is, it takes the average of two adjacent samples. Why is it a Low-Pass Filter? The running average of an input signal with little or no variation from sample to sample is very close to the input signal. Consider an input to this filter at the lowest possible frequency Hz (DC), where the signal has a single value over time: x (n) = [A, A, A,...]. The output of the lowpass filter is y(n) = x (n) + x (n ) = 2A. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 2

4 Intuitive Analyis at High Frequencies What about all the frequencies in between? The running average of an input signal with significant variation from sample to sample will be very different than the input signal. Consider a second input x 2 (n) at the highest possible frequency f s /2 (the Nyquist limit), which swings more significantly from sample to sample: x 2 (n) = [A, A, A,...]. The output of the lowpass filter is y(n) = x 2 (n) + x 2 (n ) = A A =. This filter seems to boost low frequencies while rejecting components at higher frequencies. We may find the frequency response of the filter by checking the behaviour of the filter at every possible frequency between and f s /2 Hz, a method called sinewave analysis. Alternatively, we can use an input signal that contains all of those frequencies, and then we only have to do the checking operation once. If we use an input signal with the broadest possible spectrum, i.e. an impulse, we will obtain an output from the filter, called an impulse response, which is a time domain description of the filter s response to all frequencies from DC to the Nyquist limit. The spectrum of the impulse response gives us the the frequency response of the filter. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 3 CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 4 Matlab (C-style) Low-pass Filter Implementation N = 24; % signal length x = [ zeros(, N-)]; % impulse y = [ zeros(, N-)]; % output buffer for n=2:n y(n) = x(n)+x(n-); % impulse response end Y = fft(y); % frequency response Y = abs(y(:n/2)); % amplitude response (positive frequencies) fn = [:N/2-]/N; % frequency axis subplot(2); plot(fn, Y); grid; title( Amplitude Response y(n) = x(n) + x(n-) ); xlabel( ); ylabel( ); Amplitude Response y(n) = x(n) + x(n ) Figure 5: Magnitude response for the filter y(n) = x(n) + x(n-). Filter Frequency Response Let s test the simple lowpass filter s response at frequency f by setting it s input to the complex sinusoid x(n) = Ae j(2πfnt+φ) Because of:. Time-invariance: the frequency response will not depend on φ, so we may set it to ; 2. Linearity: we may set A to. The output of the filter in response to x(n) = e j(2πfnt) is given by y(n) = x(n) + x(n ) = e jωnt + e jω(n )T = e jωnt + e jωnt e jωt = ( + e jωt )e jωnt = ( + e jωt )x(n) H(e jωt )x(n) where H(e jωt ) = ( + e jωt ), a complex multiply, is the frequency response of the filter. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 5 CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 6

5 Frequency Response Other simple non-recursive filters The filter s frequency response is acting as a complex multiply, which means a gain scaling and phase shift on the input signal. The complex filter gain in polar form is given by H(e jωt ) = G(ω)e jθ(ω) where the amplitude and phase response are given by G(ω) H(e jωt ) and Θ(ω) H(e jωt ). To solve, we may balance the exponents to obtain H(e jωt ) = ( + e jωt ) = (e jωt/2 + e jωt/2 )e jωt/2 = 2 cos(ωt/2)e jωt/2, where the amplitude response is then given by G(ω) = 2 cos(ωt/2)e jωt/2 = 2 cos(πft), f f s /2, and the phase response is given by Θ(ω) = ωt 2 = π f f s, f f s /2. Test the following filter at DC and f s /2: What type of filter is this? y(n) = x(n) x(n ). Verify that the following filter passes both DC and the Nyquist limit (/2 the sampling rate): y(n) = x(n) + x(n 2). What about at a frequency in between, f = fs/4? x(n) = [A,, A,, A,, A,...], At this frequency the filter produces no output. We may therefore assume this is a band-reject, or notch filter, with a notch at f s /4. Verify that the following filter rejects both DC and the Nyquist limit, yes boosts a frequency at f s /4. What kind of filter is it? y(n) = x(n) x(n 2). CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 7 CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 8 Plots of simple filters Generalized FIR. Amplitude Response for y(n) = x(n)+x(n ) Amplitude Response for y(n) = x(n)+x(n 2) Amplitude Response for y(n) = x(n) x(n ) Amplitude Response for y(n) = x(n) x(n 2) Figure 6: Amplitude Responses for simple filters Several different nonrecursive filters can be made by changing the delay (and thus order) and the coefficients of the filter terms. The general equation for an FIR (Finite Impulse Response) filter is given by y(n) = M b k x(n k) k= where M is the order of the filter. A filter can be defined simply by a set of coefficients. For example if b k = [, 3, 3, ] the filter is third order, (i.e. M = 3), and can be expanded into the difference equation y(n) = x(n) + 3x(n ) + 3x(n 2) + x(n 3); When the input to the FIR filter is a unit impulse sequence, δ(n) = if n = and δ(n) = otherwise, the output is the unit impulse response. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 9 CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 2

6 Matlab s filter function Increasing the Filter Order The Matlab implementation for the filter is most easily accomplished using the filter function y = filter(b, A, x). The filter function takes three (3) arguments:. feedforward coefficients B, 2. feedback coefficients A, 3. and the input signal x. If the filter doesn t have feedback coefficients, as is the case with an FIR filter, A =. Our simple low-pass filter, y(n) = x(n) + x(n ), is a first order filter with 2 feedforward coefficients (B = [, ]). In matlab: B = [ ]; A = ; y = filter(b, A, x); Let s return now to the simple low-pass filter y(n) = x(n) + x(n ) which is just a two point running average (with a factor of 2). If we increase the number of samples averaged, i.e. increase the filter order, y(n) = x(n) + x(n ) + x(n 2), the waveform will be smoothed (with a more gentle slope to zero), which corresponds to a lowered cutoff frequency Producing new filters with a cascade of simple low pass filters first order second order (scaled by /2) third order (scaled by /4) fourth order (scaled by /8) Figure 7: A Cascade of Simple Lowpass filters. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 2 CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 22 Coefficients as Impulse Response Cascade Connection If we look at the response of the filter to an impulse >> x(:) ans = >> y(:) ans = we see the impulse response y is equivalent to coefficients of our FIR filter. This can be expressed using the general FIR equation, with an input of x(n) = δ(n): h(n) = M b k δ(n k) = b n ; k= CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 23 An order can be increased by using a filter cascade. Each time the previous output y is input to the filter, a new impulse response representing is created. y(n) = x(n)+x(n ) y(n) = x(n)+x(n ) y(n) = x(n)+x(n ) y(n) = x(n)+x(n ) y(n) = x(n)+x(n ) x(n) = {,,,...} y(n) = {,,,...} y(n) = {, 2,,,,...} y(n) = {, 3, 3,,,,...} y(n) = {, 4, 6, 4,,,,...} y(n) = {, 5,,, 5,,,,...} Figure 8: A cascade of simple low-pass filters. The new output always has a finite number of non-zero components and a finite impulse response. For each filter in the cascade, the impulse response increases by one. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 24

7 Matlab LPF Cascade Implementation Recursive Filters x = [ zeros(, N-)]; B = [ ]; A = []; y = filter(b, A, x); y2 = filter(b, A, y); y3 = filter(b, A, y2); y4 = filter(b, A, y3); B = [ 5 5 ]; A = []; y = filter(b, A, x); Cascade... Or Simply... Using FIR filters often required significant computation and coefficients to reproduce a desired frequency response. It is often possible to reduce the number of feedforward coefficients needed to obtain a frequency response by introducing feedback coefficients. A simple example of a first order recursive low-pass filter is given by y(n) = b x(n) + a y(n ) The general difference equation for LTI filter therefore, is given by y(n) = b x(n) + b x(n ) + + b M x(n m) a y(n ) a N y(n N) CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 25 CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 26 FIR vs IIR Simple Low-Pass Filters Z Transforms Amplitude Response y(n) = x(n) + x(n ) Amplitude Response y(n) = x(n) + y(n ) Figure 9: Simple Non-recursive low-pass y(n) = x(n)+x(n ) (top) and Recursive low-pass y(n) = x(n) + y(n ) (bottom). The bilateral z transform of the discrete-time signal x(n) is given by X(z) x(n)z n. n= Since most signals we ll be using are causal, we will typically see the unilateral z transform given by X(z) x(n)z n. n= The z transform is a generalization of the DTFT (the DFT in the limit as the number of its samples approaches infinity). The carrier term is a generalized sampled complex sinusoid z n = e (σ+jω)t with an exponential envelope rather than a constant modulus. The DTFT equals the z transform evaluated on the unit circle in the z plane. The z transform of a signal x is given by Z{x( )} = X(z) also notated Z{x(n)} = X(z). CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 27 CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 28

8 Z Transform Properties Z Transform of the Difference Equation Two (2) important properties of z transforms:. The z transform Z{ } is a linear operator which means, by definition Z{αx (n) + βx 2 (n)} = αz{x (n)} + βz{x 2 (n)} αx (z) + βx 2 (z). 2. From the shift theorem for z transforms, the z transform of a signal delayed by M samples is given by Z{x(n M)} = z M X(z). Shift Theorem: A delay of M samples in the time domain corresponds to a multiplication by z M in the frequency domain. Recall, the general difference equation for LTI filters is given by y(n) = b x(n) + b x(n ) + + b M x(n m) a y(n ) a N y(n N). Taking the z transform of both sides yields Z{y( )} = Z{b x(n) + b x(n ) + + b M x(n m) a y(n ) a N y(n N)} Apply the linearity property to obtain Z{y( )} = b Z{x(n)} + b Z{x(n )} + + b M Z{x(n M)} a Z{y(n )} a N Z{x(n N)} Apply the shift theorem to obtain Y (z) = b X(z) + b z X(z) + + b M z M X(z) a z Y (z) a N z N Y (z) Grouping the Y (z) terms together on the left hand side and factoring out common terms X(z) and Y (z) yields Y (z)[ + a z + + a N z N ] = X(z)[b + b z + + b M z M ] CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 29 CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 3 Transfer Function More on the Transfer Function Beginning with the equation from the previous slide: Y (z)[ + a z + + a N z N ] = X(z)[b + b z + + b M z M ] We may define the following polynomials: A(z) + a z + + z N z N B(z) b + b z + + b M z M The z transform of the difference equation becomes A(z)Y (z) = B(z)X(z) In solving for Y (z)/x(z) we obtain the transfer function H(z) of a digital LTI filter H(z) = Y (z) X(z), where X(z) and Y (z) are the z transforms of the input and output signal, respectively, and H(z) is the z transform of the impulse response describing the filter. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 3 A transfer function provides an algebraic representation of a LTI filter in the frequency domain. Recall, that we can determine the frequency response by observing the effects of the filter at difference frequencies, a technique called sine-wave analysis. The gain or amplitude response of the filter at a given frequency is determined by the ratio of the the peak output amplitude to the peak input amplitude at this frequency. The phase response of the of the filter at a given frequency is determined by the difference between the output and input phases at a given frequency. The transfer function of an LTI filter is given by H(z) = Y (z) X(z) where Y (z) is the z transform of the output signal y(n) and X(z) is the z transform of the input signal x(n). The transfer function is equal to the z transform of the impulse response. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 32

9 Pole Zero Analysis Poles and Zeros cont. Recall the transfer function for the recursive LTI digital filter H(z) = g + β z + + β M z M + a z + + z N z N We can factor the numerator and denominator to obtain H(z) = g ( q z )( q 2 z ) ( q M z ) ( p z )( p 2 z ) ( p N z ) Every polynomial can be characterized by its roots plus a scale factor. We may therefore characterize a transfer function, by its. Numerator roots, called the zeros of the filter 2. Denominator roots, called poles of the filter 3. Constant gain factor The factored transfer function is given by H(z) = g ( q z )( q 2 z ) ( q M z ) ( p z )( p 2 z ) ( p N z ) Zeros The roots of the numerator polynomial are given by {q,q 2,...,q M }. When z takes on any of these values, the transfer function evaluates to zero and thus they are called the zeros of the filter. Poles The roots of the denominator polynomial are given by {p, p 2,...,p M } When z approaches any of these values, the transfer function becomes larger and larger, approaching infinity. Thus these are called the poles of the filter. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 33 CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 34 Imagining Poles and Zeros Relating Amplitude Response to Poles and Zeros Recall, the magnitude H(z) is a function of z: Since z is complex, it lies in the z plane. Since the magnitude H(z) is real, it can be represented a distance above the z plane. The plot appears as an infinitely thin surface spanning in all directions over the z plane. The zeros are the points where the surface dips down to touch the z plane. The poles look just like poles that rise forever from the z plane, getting thinner the higher they go. Notice M + feedforward coefficients gives rise to M zeros while N feedback coefficients gives rise to N poles. The filter order is given by the maximum of the numerator and denominator polynomial orders. The frequency response of the transfer function (factored form) is given by H(e jωt ) = g ( q e jωt )( q 2 e jωt ) ( q M e jωt ) ( p e jωt )( p 2 e jωt ) ( p N e jωt ) Consider the amplitude response G(ω) H(e jωt ) G(ω) = g q e jωt q 2 e jωt q M e jωt p e jωt p 2 e jωt p N e jωt = g e jmωt e jωt q e jωt q M e jnωt e jωt p e jωt p M = g ejωt q e jωt q 2 e jωt q M e jωt p e jωt p 2 e jωt p M. The amplitude response is the product of the difference between two complex numbers. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 35 CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 36

10 Difference of Two Complex Numbers Poles and Zeros and the Unit Circle In the complex plane, the number z = x + jy is plotted at the coordinates (x,y). The difference of two vectors u = x + jy and v = x 2 + jy 2 is given by u v = (x x 2 ) + j(y y 2 ). Recall the amplitude response G(ω) = g ejωt q e jωt q 2 e jωt q M e jωt p e jωt p 2 e jωt p M. imaginary imaginary u v q2 p d2 x d3 d q d4 real v u p2 x real Figure : Treatment of complex numbers as vectors in a plane. Figure : Measurement of amplitude response from a pole-zero diagram (a bi-quad section). Thus the term e jωt q i may be drawn as an arrow from the ith zero to the point e jωt on the unit circle, and e jωt p i is an arrow from the ith pole. The amplitude response at frequency ω is given by G(ω) = d d 2 d 3 d 4 CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 37 CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 38 The Biquad Section Stability The term biquad is short for bi-quadratic, and is a common name for a two-pole, two-zero digital filter. The transfer function of the biquad can be defined as H(z) = g + β z + β 2 z 2 + a z + a 2 z 2 When the coefficients a and a 2 are real (as we typically assume), the poles must be either. real (when (a /2) 2 a 2 ) 2. form a complex conjugate pair (when (a /2) 2 < a 2 ). When the poles form a complex pair, we may express them in polar form as p = Re jθ c p 2 = Re jθ c R is the pole radius, or distance from the origin in the z-plane. A filter is said to be stable if its impulse response h(n) decays to as n goes to infinity. Recall, the transfer function is the z transform of the impulse response. H(z) Z{h(n)} Consider a causal impulse response of the form h(n) = R n e jωnt,n =,, 2,... which is a damped complex sinusoid when < R < and is exponentially increasing when R >. The signal h(n) has the z transform H(z) = R n e jωnt z n = n= ( Re jωt z ) n n= = Re jωt z where the last step is a closed form representation of a geometric series and holds for Re jωt z <, which is true whenever R < z. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 39 CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 4

11 Stability cont. Bi-quadratic Resonant Filter The transfer function H(z) = has a single pole at Re jωt. Re jωt z If R > the pole of H(z) moves outside the unit circle, and the impulse response h(n) has an exponentially increasing amplitude ( h(n) = R n ). That is, it is unstable. A pole lying on the unit circle is considered marginally stable, that is, it neither decays nor grows in amplitude. In physically modelling synthesis, marginally stable poles often occur in lossless systems such as ideal vibrating strings. Therefore, to verify stability, we may find the roots of the denominator polynomial and ensure their roots are less than. For the stability of any finite order LTI filter, all its poles must lie strictly inside the unit circle. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 4 The difference for a 2 nd -order parametric bandpass filter is given by y(n) = x(n) Rx(n 2) + 2R cos(2πf c T)y(n ) R 2 y(n 2) where f c is the resonant (or center) frequency, and R, is set according to the desired bandwidth of the resonator using the following approximate relation R = e πb wt, where B w is the bandwidth at -3dB in Hz given by B w = f c Q, and T is the sampling period. This function is often called bi-quadratic or simply a biquad because both the numerator and denominator of it s transfer function are quadratic polynomials. If a resonator requires more than one mode, multiple bi-quad filters can be placed in parallel (one for each mode), with the overall output being the sum of the filter outputs. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 42 Matlab Implementation of the Biquad FIR/Convolution The two control parameters for this filter are. the center frequency f c 2. the quality factor Q (or alternatively, the bandwidth). T = /44; Q = 2; fc = 4; Bw = fc/q; R = exp(-pi*bw*t); B = [ -R]; A = [ -2*R*cos(2*pi*fc*T) R*R]; [h, w] = freqz(b,a); N = length(h); plot([:n-]/t/n/,... max(2*log(abs(h)/max(abs(h))), -4)); grid; These parameters determine the characteristic of the resonator, and can be changed in real-time, making this an efficient and ideal implementation for performance situations. Since the feedforward coefficient s of the FIR filter are the same as the non-zero elements of the impulse response, a general expression for the FIR filter s output can also be given by N y(n) = h(k)x(n k), k= where h( ) is the impulse responses and replaces the coefficients b k. When the relation between the input and the output of the FIR filter is expressed in terms of the input and impulse response, we say the the output is obtained by convolving the sequences x(n) and h(n). Note that this is circular or cyclic convolution. To simulate acyclic convolution (as we do when simulating sampled continuous-time systems), we need to zero-pad sufficiently (N + M ) so that non-zero samples do not wrap around as a resulting of shifting x. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 43 CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 44

12 Visulalizing the convolution sum Convolution The convolution sum is given by N y(n) = h(k)x(n k). k= n x(n) h(n) 3 3 h()x(n-) h()x(n-) 3 3 h(2)x(n-2) 3 3 h(3)x(n-3) y(n) The operation of convolution performed on two signals in the time domain is equivalent to the multiplication of their spectrum in the frequency domain. That is, the convolution of two time domain signals x(n) and w(n), with Fourier transforms X(ω) and W(ω) respectively, is given by Y (ω) = X(ω)W(ω) where y(n), the result of the convolution, is obtained through the inverse Fourier Transform of Y (ω). CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 45 CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 46 First-Order Shelf Filter The first-order shelf filter is specified by a transition frequency f t and a band-edge gain g π (at the Nyquist limit). The filter has unity gain at DC, and a gain of g π at the transition frequency. The transfer function of the first-order shelf filter is given by H(z;f t,g π ) = b + b z + a z, with coefficients given by b = β + ρβ, + ρα b = β + ρβ, + ρα a = ρ + α, + ρα where and and β = ( + g π )/2 + ( g π )α /2, β = ( g π )/2 + ( g + π)α /2,, g π = α = η sign(η)(η 2 ) 2, g π. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 47 To form the coefficients of a prototype shelf filter with a transition frequency of π/2, η is set to the following: The parameter η = (g π + )/(g π ), g π. ρ = sin(πf t /2 π/4)/ sin(πf t /2 + π/4) is the coefficient of the first-order allpass transformation warping the prototype filter to the proper transition frequency. CMPT 889: Computational Modelling for Sound Synthesis: Lecture 5 48

CMPT 889: Lecture 5 Filters

CMPT 889: Lecture 5 Filters CMPT 889: Lecture 5 Filters Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University October 7, 2009 1 Digital Filters Any medium through which a signal passes may be regarded

More information

Lecture 19 IIR Filters

Lecture 19 IIR Filters Lecture 19 IIR Filters Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/5/10 1 General IIR Difference Equation IIR system: infinite-impulse response system The most general class

More information

CMPT 318: Lecture 5 Complex Exponentials, Spectrum Representation

CMPT 318: Lecture 5 Complex Exponentials, Spectrum Representation CMPT 318: Lecture 5 Complex Exponentials, Spectrum Representation Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University January 23, 2006 1 Exponentials The exponential is

More information

DIGITAL SIGNAL PROCESSING UNIT III INFINITE IMPULSE RESPONSE DIGITAL FILTERS. 3.6 Design of Digital Filter using Digital to Digital

DIGITAL SIGNAL PROCESSING UNIT III INFINITE IMPULSE RESPONSE DIGITAL FILTERS. 3.6 Design of Digital Filter using Digital to Digital DIGITAL SIGNAL PROCESSING UNIT III INFINITE IMPULSE RESPONSE DIGITAL FILTERS Contents: 3.1 Introduction IIR Filters 3.2 Transformation Function Derivation 3.3 Review of Analog IIR Filters 3.3.1 Butterworth

More information

Discrete Time Systems

Discrete Time Systems Discrete Time Systems Valentina Hubeika, Jan Černocký DCGM FIT BUT Brno, {ihubeika,cernocky}@fit.vutbr.cz 1 LTI systems In this course, we work only with linear and time-invariant systems. We talked about

More information

Ch. 7: Z-transform Reading

Ch. 7: Z-transform Reading c J. Fessler, June 9, 3, 6:3 (student version) 7. Ch. 7: Z-transform Definition Properties linearity / superposition time shift convolution: y[n] =h[n] x[n] Y (z) =H(z) X(z) Inverse z-transform by coefficient

More information

Discrete Time Systems

Discrete Time Systems 1 Discrete Time Systems {x[0], x[1], x[2], } H {y[0], y[1], y[2], } Example: y[n] = 2x[n] + 3x[n-1] + 4x[n-2] 2 FIR and IIR Systems FIR: Finite Impulse Response -- non-recursive y[n] = 2x[n] + 3x[n-1]

More information

Stability Condition in Terms of the Pole Locations

Stability Condition in Terms of the Pole Locations Stability Condition in Terms of the Pole Locations A causal LTI digital filter is BIBO stable if and only if its impulse response h[n] is absolutely summable, i.e., 1 = S h [ n] < n= We now develop a stability

More information

Responses of Digital Filters Chapter Intended Learning Outcomes:

Responses of Digital Filters Chapter Intended Learning Outcomes: Responses of Digital Filters Chapter Intended Learning Outcomes: (i) Understanding the relationships between impulse response, frequency response, difference equation and transfer function in characterizing

More information

UNIVERSITY OF OSLO. Please make sure that your copy of the problem set is complete before you attempt to answer anything.

UNIVERSITY OF OSLO. Please make sure that your copy of the problem set is complete before you attempt to answer anything. UNIVERSITY OF OSLO Faculty of mathematics and natural sciences Examination in INF3470/4470 Digital signal processing Day of examination: December 9th, 011 Examination hours: 14.30 18.30 This problem set

More information

# FIR. [ ] = b k. # [ ]x[ n " k] [ ] = h k. x[ n] = Ae j" e j# ˆ n Complex exponential input. [ ]Ae j" e j ˆ. ˆ )Ae j# e j ˆ. y n. y n.

# FIR. [ ] = b k. # [ ]x[ n  k] [ ] = h k. x[ n] = Ae j e j# ˆ n Complex exponential input. [ ]Ae j e j ˆ. ˆ )Ae j# e j ˆ. y n. y n. [ ] = h k M [ ] = b k x[ n " k] FIR k= M [ ]x[ n " k] convolution k= x[ n] = Ae j" e j ˆ n Complex exponential input [ ] = h k M % k= [ ]Ae j" e j ˆ % M = ' h[ k]e " j ˆ & k= k = H (" ˆ )Ae j e j ˆ ( )

More information

Review of Discrete-Time System

Review of Discrete-Time System Review of Discrete-Time System Electrical & Computer Engineering University of Maryland, College Park Acknowledgment: ENEE630 slides were based on class notes developed by Profs. K.J. Ray Liu and Min Wu.

More information

Music 270a: Complex Exponentials and Spectrum Representation

Music 270a: Complex Exponentials and Spectrum Representation Music 270a: Complex Exponentials and Spectrum Representation Tamara Smyth, trsmyth@ucsd.edu Department of Music, University of California, San Diego (UCSD) October 24, 2016 1 Exponentials The exponential

More information

Sinusoids. Amplitude and Magnitude. Phase and Period. CMPT 889: Lecture 2 Sinusoids, Complex Exponentials, Spectrum Representation

Sinusoids. Amplitude and Magnitude. Phase and Period. CMPT 889: Lecture 2 Sinusoids, Complex Exponentials, Spectrum Representation Sinusoids CMPT 889: Lecture Sinusoids, Complex Exponentials, Spectrum Representation Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University September 6, 005 Sinusoids are

More information

E : Lecture 1 Introduction

E : Lecture 1 Introduction E85.2607: Lecture 1 Introduction 1 Administrivia 2 DSP review 3 Fun with Matlab E85.2607: Lecture 1 Introduction 2010-01-21 1 / 24 Course overview Advanced Digital Signal Theory Design, analysis, and implementation

More information

Very useful for designing and analyzing signal processing systems

Very useful for designing and analyzing signal processing systems z-transform z-transform The z-transform generalizes the Discrete-Time Fourier Transform (DTFT) for analyzing infinite-length signals and systems Very useful for designing and analyzing signal processing

More information

CMPT 889: Lecture 2 Sinusoids, Complex Exponentials, Spectrum Representation

CMPT 889: Lecture 2 Sinusoids, Complex Exponentials, Spectrum Representation CMPT 889: Lecture 2 Sinusoids, Complex Exponentials, Spectrum Representation Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University September 26, 2005 1 Sinusoids Sinusoids

More information

EECE 301 Signals & Systems Prof. Mark Fowler

EECE 301 Signals & Systems Prof. Mark Fowler EECE 3 Signals & Systems Prof. ark Fowler Note Set #28 D-T Systems: DT Filters Ideal & Practical /4 Ideal D-T Filters Just as in the CT case we can specify filters. We looked at the ideal filter for the

More information

Discrete-time signals and systems

Discrete-time signals and systems Discrete-time signals and systems 1 DISCRETE-TIME DYNAMICAL SYSTEMS x(t) G y(t) Linear system: Output y(n) is a linear function of the inputs sequence: y(n) = k= h(k)x(n k) h(k): impulse response of the

More information

Lecture Notes on Fundamentals of Audio-Video Signal Processing

Lecture Notes on Fundamentals of Audio-Video Signal Processing Marco Tagliasacchi Lecture Notes on Fundamentals of Audio-Video Signal Processing November 12, 2007 Politecnico di Milano Contents Part I Fundamentals of Digital Signal Processing 1 Introduction to the

More information

Like bilateral Laplace transforms, ROC must be used to determine a unique inverse z-transform.

Like bilateral Laplace transforms, ROC must be used to determine a unique inverse z-transform. Inversion of the z-transform Focus on rational z-transform of z 1. Apply partial fraction expansion. Like bilateral Laplace transforms, ROC must be used to determine a unique inverse z-transform. Let X(z)

More information

EE482: Digital Signal Processing Applications

EE482: Digital Signal Processing Applications Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu EE482: Digital Signal Processing Applications Spring 2014 TTh 14:30-15:45 CBC C222 Lecture 02 DSP Fundamentals 14/01/21 http://www.ee.unlv.edu/~b1morris/ee482/

More information

EC Signals and Systems

EC Signals and Systems UNIT I CLASSIFICATION OF SIGNALS AND SYSTEMS Continuous time signals (CT signals), discrete time signals (DT signals) Step, Ramp, Pulse, Impulse, Exponential 1. Define Unit Impulse Signal [M/J 1], [M/J

More information

Review of Fundamentals of Digital Signal Processing

Review of Fundamentals of Digital Signal Processing Chapter 2 Review of Fundamentals of Digital Signal Processing 2.1 (a) This system is not linear (the constant term makes it non linear) but is shift-invariant (b) This system is linear but not shift-invariant

More information

CMPT 889: Lecture 9 Wind Instrument Modelling

CMPT 889: Lecture 9 Wind Instrument Modelling CMPT 889: Lecture 9 Wind Instrument Modelling Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University November 20, 2006 1 Scattering Scattering is a phenomenon in which the

More information

Damped Oscillators (revisited)

Damped Oscillators (revisited) Damped Oscillators (revisited) We saw that damped oscillators can be modeled using a recursive filter with two coefficients and no feedforward components: Y(k) = - a(1)*y(k-1) - a(2)*y(k-2) We derived

More information

Review of Fundamentals of Digital Signal Processing

Review of Fundamentals of Digital Signal Processing Solution Manual for Theory and Applications of Digital Speech Processing by Lawrence Rabiner and Ronald Schafer Click here to Purchase full Solution Manual at http://solutionmanuals.info Link download

More information

INFINITE-IMPULSE RESPONSE DIGITAL FILTERS Classical analog filters and their conversion to digital filters 4. THE BUTTERWORTH ANALOG FILTER

INFINITE-IMPULSE RESPONSE DIGITAL FILTERS Classical analog filters and their conversion to digital filters 4. THE BUTTERWORTH ANALOG FILTER INFINITE-IMPULSE RESPONSE DIGITAL FILTERS Classical analog filters and their conversion to digital filters. INTRODUCTION 2. IIR FILTER DESIGN 3. ANALOG FILTERS 4. THE BUTTERWORTH ANALOG FILTER 5. THE CHEBYSHEV-I

More information

PS403 - Digital Signal processing

PS403 - Digital Signal processing PS403 - Digital Signal processing 6. DSP - Recursive (IIR) Digital Filters Key Text: Digital Signal Processing with Computer Applications (2 nd Ed.) Paul A Lynn and Wolfgang Fuerst, (Publisher: John Wiley

More information

Z-Transform. x (n) Sampler

Z-Transform. x (n) Sampler Chapter Two A- Discrete Time Signals: The discrete time signal x(n) is obtained by taking samples of the analog signal xa (t) every Ts seconds as shown in Figure below. Analog signal Discrete time signal

More information

Introduction to DSP Time Domain Representation of Signals and Systems

Introduction to DSP Time Domain Representation of Signals and Systems Introduction to DSP Time Domain Representation of Signals and Systems Dr. Waleed Al-Hanafy waleed alhanafy@yahoo.com Faculty of Electronic Engineering, Menoufia Univ., Egypt Digital Signal Processing (ECE407)

More information

How to manipulate Frequencies in Discrete-time Domain? Two Main Approaches

How to manipulate Frequencies in Discrete-time Domain? Two Main Approaches How to manipulate Frequencies in Discrete-time Domain? Two Main Approaches Difference Equations (an LTI system) x[n]: input, y[n]: output That is, building a system that maes use of the current and previous

More information

Let H(z) = P(z)/Q(z) be the system function of a rational form. Let us represent both P(z) and Q(z) as polynomials of z (not z -1 )

Let H(z) = P(z)/Q(z) be the system function of a rational form. Let us represent both P(z) and Q(z) as polynomials of z (not z -1 ) Review: Poles and Zeros of Fractional Form Let H() = P()/Q() be the system function of a rational form. Let us represent both P() and Q() as polynomials of (not - ) Then Poles: the roots of Q()=0 Zeros:

More information

ECE503: Digital Signal Processing Lecture 5

ECE503: Digital Signal Processing Lecture 5 ECE53: Digital Signal Processing Lecture 5 D. Richard Brown III WPI 3-February-22 WPI D. Richard Brown III 3-February-22 / 32 Lecture 5 Topics. Magnitude and phase characterization of transfer functions

More information

Linear Convolution Using FFT

Linear Convolution Using FFT Linear Convolution Using FFT Another useful property is that we can perform circular convolution and see how many points remain the same as those of linear convolution. When P < L and an L-point circular

More information

Transform analysis of LTI systems Oppenheim and Schafer, Second edition pp For LTI systems we can write

Transform analysis of LTI systems Oppenheim and Schafer, Second edition pp For LTI systems we can write Transform analysis of LTI systems Oppenheim and Schafer, Second edition pp. 4 9. For LTI systems we can write yœn D xœn hœn D X kd xœkhœn Alternatively, this relationship can be expressed in the z-transform

More information

Digital Signal Processing Lecture 8 - Filter Design - IIR

Digital Signal Processing Lecture 8 - Filter Design - IIR Digital Signal Processing - Filter Design - IIR Electrical Engineering and Computer Science University of Tennessee, Knoxville October 20, 2015 Overview 1 2 3 4 5 6 Roadmap Discrete-time signals and systems

More information

Digital Signal Processing Lecture 9 - Design of Digital Filters - FIR

Digital Signal Processing Lecture 9 - Design of Digital Filters - FIR Digital Signal Processing - Design of Digital Filters - FIR Electrical Engineering and Computer Science University of Tennessee, Knoxville November 3, 2015 Overview 1 2 3 4 Roadmap Introduction Discrete-time

More information

Lecture 7 Discrete Systems

Lecture 7 Discrete Systems Lecture 7 Discrete Systems EE 52: Instrumentation and Measurements Lecture Notes Update on November, 29 Aly El-Osery, Electrical Engineering Dept., New Mexico Tech 7. Contents The z-transform 2 Linear

More information

/ (2π) X(e jω ) dω. 4. An 8 point sequence is given by x(n) = {2,2,2,2,1,1,1,1}. Compute 8 point DFT of x(n) by

/ (2π) X(e jω ) dω. 4. An 8 point sequence is given by x(n) = {2,2,2,2,1,1,1,1}. Compute 8 point DFT of x(n) by Code No: RR320402 Set No. 1 III B.Tech II Semester Regular Examinations, Apr/May 2006 DIGITAL SIGNAL PROCESSING ( Common to Electronics & Communication Engineering, Electronics & Instrumentation Engineering,

More information

EE 521: Instrumentation and Measurements

EE 521: Instrumentation and Measurements Aly El-Osery Electrical Engineering Department, New Mexico Tech Socorro, New Mexico, USA November 1, 2009 1 / 27 1 The z-transform 2 Linear Time-Invariant System 3 Filter Design IIR Filters FIR Filters

More information

Analog vs. discrete signals

Analog vs. discrete signals Analog vs. discrete signals Continuous-time signals are also known as analog signals because their amplitude is analogous (i.e., proportional) to the physical quantity they represent. Discrete-time signals

More information

Digital Filters Ying Sun

Digital Filters Ying Sun Digital Filters Ying Sun Digital filters Finite impulse response (FIR filter: h[n] has a finite numbers of terms. Infinite impulse response (IIR filter: h[n] has infinite numbers of terms. Causal filter:

More information

Scattering. N-port Parallel Scattering Junctions. Physical Variables at the Junction. CMPT 889: Lecture 9 Wind Instrument Modelling

Scattering. N-port Parallel Scattering Junctions. Physical Variables at the Junction. CMPT 889: Lecture 9 Wind Instrument Modelling Scattering CMPT 889: Lecture 9 Wind Instrument Modelling Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser niversity November 2, 26 Scattering is a phenomenon in which the wave

More information

Convolution. Define a mathematical operation on discrete-time signals called convolution, represented by *. Given two discrete-time signals x 1, x 2,

Convolution. Define a mathematical operation on discrete-time signals called convolution, represented by *. Given two discrete-time signals x 1, x 2, Filters Filters So far: Sound signals, connection to Fourier Series, Introduction to Fourier Series and Transforms, Introduction to the FFT Today Filters Filters: Keep part of the signal we are interested

More information

DHANALAKSHMI COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING EC2314- DIGITAL SIGNAL PROCESSING UNIT I INTRODUCTION PART A

DHANALAKSHMI COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING EC2314- DIGITAL SIGNAL PROCESSING UNIT I INTRODUCTION PART A DHANALAKSHMI COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING EC2314- DIGITAL SIGNAL PROCESSING UNIT I INTRODUCTION PART A Classification of systems : Continuous and Discrete

More information

Fourier Series Representation of

Fourier Series Representation of Fourier Series Representation of Periodic Signals Rui Wang, Assistant professor Dept. of Information and Communication Tongji University it Email: ruiwang@tongji.edu.cn Outline The response of LIT system

More information

z-transforms Definition of the z-transform Chapter

z-transforms Definition of the z-transform Chapter z-transforms Chapter 7 In the study of discrete-time signal and systems, we have thus far considered the time-domain and the frequency domain. The z- domain gives us a third representation. All three domains

More information

Signals and Systems. Lecture 11 Wednesday 22 nd November 2017 DR TANIA STATHAKI

Signals and Systems. Lecture 11 Wednesday 22 nd November 2017 DR TANIA STATHAKI Signals and Systems Lecture 11 Wednesday 22 nd November 2017 DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE LONDON Effect on poles and zeros on frequency response

More information

Digital Signal Processing

Digital Signal Processing COMP ENG 4TL4: Digital Signal Processing Notes for Lecture #21 Friday, October 24, 2003 Types of causal FIR (generalized) linear-phase filters: Type I: Symmetric impulse response: with order M an even

More information

The Discrete-Time Fourier

The Discrete-Time Fourier Chapter 3 The Discrete-Time Fourier Transform 清大電機系林嘉文 cwlin@ee.nthu.edu.tw 03-5731152 Original PowerPoint slides prepared by S. K. Mitra 3-1-1 Continuous-Time Fourier Transform Definition The CTFT of

More information

Lecture 11 FIR Filters

Lecture 11 FIR Filters Lecture 11 FIR Filters Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/4/12 1 The Unit Impulse Sequence Any sequence can be represented in this way. The equation is true if k ranges

More information

Review of Linear Time-Invariant Network Analysis

Review of Linear Time-Invariant Network Analysis D1 APPENDIX D Review of Linear Time-Invariant Network Analysis Consider a network with input x(t) and output y(t) as shown in Figure D-1. If an input x 1 (t) produces an output y 1 (t), and an input x

More information

IT DIGITAL SIGNAL PROCESSING (2013 regulation) UNIT-1 SIGNALS AND SYSTEMS PART-A

IT DIGITAL SIGNAL PROCESSING (2013 regulation) UNIT-1 SIGNALS AND SYSTEMS PART-A DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING IT6502 - DIGITAL SIGNAL PROCESSING (2013 regulation) UNIT-1 SIGNALS AND SYSTEMS PART-A 1. What is a continuous and discrete time signal? Continuous

More information

Digital Signal Processing Lecture 10 - Discrete Fourier Transform

Digital Signal Processing Lecture 10 - Discrete Fourier Transform Digital Signal Processing - Discrete Fourier Transform Electrical Engineering and Computer Science University of Tennessee, Knoxville November 12, 2015 Overview 1 2 3 4 Review - 1 Introduction Discrete-time

More information

Digital Signal Processing Lecture 4

Digital Signal Processing Lecture 4 Remote Sensing Laboratory Dept. of Information Engineering and Computer Science University of Trento Via Sommarive, 14, I-38123 Povo, Trento, Italy Digital Signal Processing Lecture 4 Begüm Demir E-mail:

More information

UNIVERSITY OF OSLO. Faculty of mathematics and natural sciences. Forslag til fasit, versjon-01: Problem 1 Signals and systems.

UNIVERSITY OF OSLO. Faculty of mathematics and natural sciences. Forslag til fasit, versjon-01: Problem 1 Signals and systems. UNIVERSITY OF OSLO Faculty of mathematics and natural sciences Examination in INF3470/4470 Digital signal processing Day of examination: December 1th, 016 Examination hours: 14:30 18.30 This problem set

More information

Universiti Malaysia Perlis EKT430: DIGITAL SIGNAL PROCESSING LAB ASSIGNMENT 3: DISCRETE TIME SYSTEM IN TIME DOMAIN

Universiti Malaysia Perlis EKT430: DIGITAL SIGNAL PROCESSING LAB ASSIGNMENT 3: DISCRETE TIME SYSTEM IN TIME DOMAIN Universiti Malaysia Perlis EKT430: DIGITAL SIGNAL PROCESSING LAB ASSIGNMENT 3: DISCRETE TIME SYSTEM IN TIME DOMAIN Pusat Pengajian Kejuruteraan Komputer Dan Perhubungan Universiti Malaysia Perlis Discrete-Time

More information

Z-TRANSFORMS. Solution: Using the definition (5.1.2), we find: for case (b). y(n)= h(n) x(n) Y(z)= H(z)X(z) (convolution) (5.1.

Z-TRANSFORMS. Solution: Using the definition (5.1.2), we find: for case (b). y(n)= h(n) x(n) Y(z)= H(z)X(z) (convolution) (5.1. 84 5. Z-TRANSFORMS 5 z-transforms Solution: Using the definition (5..2), we find: for case (a), and H(z) h 0 + h z + h 2 z 2 + h 3 z 3 2 + 3z + 5z 2 + 2z 3 H(z) h 0 + h z + h 2 z 2 + h 3 z 3 + h 4 z 4

More information

Use: Analysis of systems, simple convolution, shorthand for e jw, stability. Motivation easier to write. Or X(z) = Z {x(n)}

Use: Analysis of systems, simple convolution, shorthand for e jw, stability. Motivation easier to write. Or X(z) = Z {x(n)} 1 VI. Z Transform Ch 24 Use: Analysis of systems, simple convolution, shorthand for e jw, stability. A. Definition: X(z) = x(n) z z - transforms Motivation easier to write Or Note if X(z) = Z {x(n)} z

More information

Signals and Systems. Problem Set: The z-transform and DT Fourier Transform

Signals and Systems. Problem Set: The z-transform and DT Fourier Transform Signals and Systems Problem Set: The z-transform and DT Fourier Transform Updated: October 9, 7 Problem Set Problem - Transfer functions in MATLAB A discrete-time, causal LTI system is described by the

More information

CMPT 889: Lecture 3 Fundamentals of Digital Audio, Discrete-Time Signals

CMPT 889: Lecture 3 Fundamentals of Digital Audio, Discrete-Time Signals CMPT 889: Lecture 3 Fundamentals of Digital Audio, Discrete-Time Signals Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University October 6, 2005 1 Sound Sound waves are longitudinal

More information

GATE EE Topic wise Questions SIGNALS & SYSTEMS

GATE EE Topic wise Questions SIGNALS & SYSTEMS www.gatehelp.com GATE EE Topic wise Questions YEAR 010 ONE MARK Question. 1 For the system /( s + 1), the approximate time taken for a step response to reach 98% of the final value is (A) 1 s (B) s (C)

More information

Lecture 04: Discrete Frequency Domain Analysis (z-transform)

Lecture 04: Discrete Frequency Domain Analysis (z-transform) Lecture 04: Discrete Frequency Domain Analysis (z-transform) John Chiverton School of Information Technology Mae Fah Luang University 1st Semester 2009/ 2552 Outline Overview Lecture Contents Introduction

More information

Lecture 27 Frequency Response 2

Lecture 27 Frequency Response 2 Lecture 27 Frequency Response 2 Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/6/12 1 Application of Ideal Filters Suppose we can generate a square wave with a fundamental period

More information

Signals and Systems. Spring Room 324, Geology Palace, ,

Signals and Systems. Spring Room 324, Geology Palace, , Signals and Systems Spring 2013 Room 324, Geology Palace, 13756569051, zhukaiguang@jlu.edu.cn Chapter 10 The Z-Transform 1) Z-Transform 2) Properties of the ROC of the z-transform 3) Inverse z-transform

More information

Signals & Systems Handout #4

Signals & Systems Handout #4 Signals & Systems Handout #4 H-4. Elementary Discrete-Domain Functions (Sequences): Discrete-domain functions are defined for n Z. H-4.. Sequence Notation: We use the following notation to indicate the

More information

Chapter 7: Filter Design 7.1 Practical Filter Terminology

Chapter 7: Filter Design 7.1 Practical Filter Terminology hapter 7: Filter Design 7. Practical Filter Terminology Analog and digital filters and their designs constitute one of the major emphasis areas in signal processing and communication systems. This is due

More information

Introduction to Digital Signal Processing

Introduction to Digital Signal Processing Introduction to Digital Signal Processing 1.1 What is DSP? DSP is a technique of performing the mathematical operations on the signals in digital domain. As real time signals are analog in nature we need

More information

EEL3135: Homework #4

EEL3135: Homework #4 EEL335: Homework #4 Problem : For each of the systems below, determine whether or not the system is () linear, () time-invariant, and (3) causal: (a) (b) (c) xn [ ] cos( 04πn) (d) xn [ ] xn [ ] xn [ 5]

More information

CHAPTER 2 RANDOM PROCESSES IN DISCRETE TIME

CHAPTER 2 RANDOM PROCESSES IN DISCRETE TIME CHAPTER 2 RANDOM PROCESSES IN DISCRETE TIME Shri Mata Vaishno Devi University, (SMVDU), 2013 Page 13 CHAPTER 2 RANDOM PROCESSES IN DISCRETE TIME When characterizing or modeling a random variable, estimates

More information

Discrete-Time Fourier Transform (DTFT)

Discrete-Time Fourier Transform (DTFT) Discrete-Time Fourier Transform (DTFT) 1 Preliminaries Definition: The Discrete-Time Fourier Transform (DTFT) of a signal x[n] is defined to be X(e jω ) x[n]e jωn. (1) In other words, the DTFT of x[n]

More information

Multimedia Signals and Systems - Audio and Video. Signal, Image, Video Processing Review-Introduction, MP3 and MPEG2

Multimedia Signals and Systems - Audio and Video. Signal, Image, Video Processing Review-Introduction, MP3 and MPEG2 Multimedia Signals and Systems - Audio and Video Signal, Image, Video Processing Review-Introduction, MP3 and MPEG2 Kunio Takaya Electrical and Computer Engineering University of Saskatchewan December

More information

Discrete-Time Signals and Systems. Frequency Domain Analysis of LTI Systems. The Frequency Response Function. The Frequency Response Function

Discrete-Time Signals and Systems. Frequency Domain Analysis of LTI Systems. The Frequency Response Function. The Frequency Response Function Discrete-Time Signals and s Frequency Domain Analysis of LTI s Dr. Deepa Kundur University of Toronto Reference: Sections 5., 5.2-5.5 of John G. Proakis and Dimitris G. Manolakis, Digital Signal Processing:

More information

LAB 6: FIR Filter Design Summer 2011

LAB 6: FIR Filter Design Summer 2011 University of Illinois at Urbana-Champaign Department of Electrical and Computer Engineering ECE 311: Digital Signal Processing Lab Chandra Radhakrishnan Peter Kairouz LAB 6: FIR Filter Design Summer 011

More information

Discrete-Time Fourier Transform

Discrete-Time Fourier Transform C H A P T E R 7 Discrete-Time Fourier Transform In Chapter 3 and Appendix C, we showed that interesting continuous-time waveforms x(t) can be synthesized by summing sinusoids, or complex exponential signals,

More information

A system that is both linear and time-invariant is called linear time-invariant (LTI).

A system that is both linear and time-invariant is called linear time-invariant (LTI). The Cooper Union Department of Electrical Engineering ECE111 Signal Processing & Systems Analysis Lecture Notes: Time, Frequency & Transform Domains February 28, 2012 Signals & Systems Signals are mapped

More information

Z Transform (Part - II)

Z Transform (Part - II) Z Transform (Part - II). The Z Transform of the following real exponential sequence x(nt) = a n, nt 0 = 0, nt < 0, a > 0 (a) ; z > (c) for all z z (b) ; z (d) ; z < a > a az az Soln. The given sequence

More information

ECE-314 Fall 2012 Review Questions for Midterm Examination II

ECE-314 Fall 2012 Review Questions for Midterm Examination II ECE-314 Fall 2012 Review Questions for Midterm Examination II First, make sure you study all the problems and their solutions from homework sets 4-7. Then work on the following additional problems. Problem

More information

Discrete Time Signals and Systems Time-frequency Analysis. Gloria Menegaz

Discrete Time Signals and Systems Time-frequency Analysis. Gloria Menegaz Discrete Time Signals and Systems Time-frequency Analysis Gloria Menegaz Time-frequency Analysis Fourier transform (1D and 2D) Reference textbook: Discrete time signal processing, A.W. Oppenheim and R.W.

More information

Your solutions for time-domain waveforms should all be expressed as real-valued functions.

Your solutions for time-domain waveforms should all be expressed as real-valued functions. ECE-486 Test 2, Feb 23, 2017 2 Hours; Closed book; Allowed calculator models: (a) Casio fx-115 models (b) HP33s and HP 35s (c) TI-30X and TI-36X models. Calculators not included in this list are not permitted.

More information

Lecture 3 - Design of Digital Filters

Lecture 3 - Design of Digital Filters Lecture 3 - Design of Digital Filters 3.1 Simple filters In the previous lecture we considered the polynomial fit as a case example of designing a smoothing filter. The approximation to an ideal LPF can

More information

ELEG 305: Digital Signal Processing

ELEG 305: Digital Signal Processing ELEG 305: Digital Signal Processing Lecture : Design of Digital IIR Filters (Part I) Kenneth E. Barner Department of Electrical and Computer Engineering University of Delaware Fall 008 K. E. Barner (Univ.

More information

VI. Z Transform and DT System Analysis

VI. Z Transform and DT System Analysis Summer 2008 Signals & Systems S.F. Hsieh VI. Z Transform and DT System Analysis Introduction Why Z transform? a DT counterpart of the Laplace transform in CT. Generalization of DT Fourier transform: z

More information

NAME: 11 December 2013 Digital Signal Processing I Final Exam Fall Cover Sheet

NAME: 11 December 2013 Digital Signal Processing I Final Exam Fall Cover Sheet NAME: December Digital Signal Processing I Final Exam Fall Cover Sheet Test Duration: minutes. Open Book but Closed Notes. Three 8.5 x crib sheets allowed Calculators NOT allowed. This test contains four

More information

Question Paper Code : AEC11T02

Question Paper Code : AEC11T02 Hall Ticket No Question Paper Code : AEC11T02 VARDHAMAN COLLEGE OF ENGINEERING (AUTONOMOUS) Affiliated to JNTUH, Hyderabad Four Year B. Tech III Semester Tutorial Question Bank 2013-14 (Regulations: VCE-R11)

More information

Lecture 18: Stability

Lecture 18: Stability Lecture 18: Stability ECE 401: Signal and Image Analysis University of Illinois 4/18/2017 1 Stability 2 Impulse Response 3 Z Transform Outline 1 Stability 2 Impulse Response 3 Z Transform BIBO Stability

More information

(i) Represent discrete-time signals using transform. (ii) Understand the relationship between transform and discrete-time Fourier transform

(i) Represent discrete-time signals using transform. (ii) Understand the relationship between transform and discrete-time Fourier transform z Transform Chapter Intended Learning Outcomes: (i) Represent discrete-time signals using transform (ii) Understand the relationship between transform and discrete-time Fourier transform (iii) Understand

More information

DSP-I DSP-I DSP-I DSP-I

DSP-I DSP-I DSP-I DSP-I DSP-I DSP-I DSP-I DSP-I Digital Signal Processing I (8-79) Fall Semester, 005 OTES FOR 8-79 LECTURE 9: PROPERTIES AD EXAPLES OF Z-TRASFORS Distributed: September 7, 005 otes: This handout contains in outline

More information

Lecture 2 OKAN UNIVERSITY FACULTY OF ENGINEERING AND ARCHITECTURE

Lecture 2 OKAN UNIVERSITY FACULTY OF ENGINEERING AND ARCHITECTURE OKAN UNIVERSITY FACULTY OF ENGINEERING AND ARCHITECTURE EEE 43 DIGITAL SIGNAL PROCESSING (DSP) 2 DIFFERENCE EQUATIONS AND THE Z- TRANSFORM FALL 22 Yrd. Doç. Dr. Didem Kivanc Tureli didemk@ieee.org didem.kivanc@okan.edu.tr

More information

Chapter Intended Learning Outcomes: (i) Understanding the relationship between transform and the Fourier transform for discrete-time signals

Chapter Intended Learning Outcomes: (i) Understanding the relationship between transform and the Fourier transform for discrete-time signals z Transform Chapter Intended Learning Outcomes: (i) Understanding the relationship between transform and the Fourier transform for discrete-time signals (ii) Understanding the characteristics and properties

More information

ECE-S Introduction to Digital Signal Processing Lecture 4 Part A The Z-Transform and LTI Systems

ECE-S Introduction to Digital Signal Processing Lecture 4 Part A The Z-Transform and LTI Systems ECE-S352-70 Introduction to Digital Signal Processing Lecture 4 Part A The Z-Transform and LTI Systems Transform techniques are an important tool in the analysis of signals and linear time invariant (LTI)

More information

UNIT 4: DIGITAL SYSTEM MODELS

UNIT 4: DIGITAL SYSTEM MODELS UNIT 4: DIGITAL SYSTEM MODELS 4.1 Introduction This unit is concerned with the description of digital systems, it introduces the concepts of a linear time-invariant system, convolution, the general system

More information

Module 4 : Laplace and Z Transform Problem Set 4

Module 4 : Laplace and Z Transform Problem Set 4 Module 4 : Laplace and Z Transform Problem Set 4 Problem 1 The input x(t) and output y(t) of a causal LTI system are related to the block diagram representation shown in the figure. (a) Determine a differential

More information

Discrete-Time David Johns and Ken Martin University of Toronto

Discrete-Time David Johns and Ken Martin University of Toronto Discrete-Time David Johns and Ken Martin University of Toronto (johns@eecg.toronto.edu) (martin@eecg.toronto.edu) University of Toronto 1 of 40 Overview of Some Signal Spectra x c () t st () x s () t xn

More information

Poles and Zeros in z-plane

Poles and Zeros in z-plane M58 Mixed Signal Processors page of 6 Poles and Zeros in z-plane z-plane Response of discrete-time system (i.e. digital filter at a particular frequency ω is determined by the distance between its poles

More information

Matched Second Order Digital Filters

Matched Second Order Digital Filters Matched Second Order Digital Filters Martin Vicanek 14. February 016 1 Introduction Second order sections are universal building blocks for digital filters. They are characterized by five coefficients,

More information

Multidimensional digital signal processing

Multidimensional digital signal processing PSfrag replacements Two-dimensional discrete signals N 1 A 2-D discrete signal (also N called a sequence or array) is a function 2 defined over thex(n set 1 of, n 2 ordered ) pairs of integers: y(nx 1,

More information

Digital Signal Processing Lecture 5

Digital Signal Processing Lecture 5 Remote Sensing Laboratory Dept. of Information Engineering and Computer Science University of Trento Via Sommarive, 14, I-38123 Povo, Trento, Italy Digital Signal Processing Lecture 5 Begüm Demir E-mail:

More information

Problem Set 9 Solutions

Problem Set 9 Solutions Problem Set 9 Solutions EE23: Digital Signal Processing. From Figure below, we see that the DTFT of the windowed sequence approaches the actual DTFT as the window size increases. Gibb s phenomenon is absent

More information