An Fir-Filter Example: Hanning Filter
|
|
- Nicholas Wiggins
- 6 years ago
- Views:
Transcription
1 An Fir-Filter Example: Hanning Filter Josef Goette Bern University of Applied Sciences, Biel Institute of Human Centered Engineering - microlab Josef.Goette@bfh.ch February 7, 2018 Contents 1 Mathematical Descriptions Transfer Function Frequency Response Poles and Zeros Signal Flow Graphs Direct Form Transposed Form Cascade Form References Fir Hanning i 2018
2 c Josef Goette, All rights reserved. This work may not be translated or copied in whole or in part without the written permission by the author, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software is forbidden. 31 Fir Hanning ii 2018
3 1 Mathematical Descriptions The filter might be described in the time domain as well as in transformation domains. Whereas the difference equation in the time domain is the natural description, which is most often used in implementations, the transformation domain descriptions are useful for the purpose of analysis and synthesis of the filter. As transformation-domain descriptions we use the domain of the z-transform leading to the transfer function, and the frequencydomain description leading to the frequency response. Time- and Transform-Domain Descriptions difference equation y[n] = 1 4 ( ) x[n] + 2x[n 1] + x[n 2] transfer function H(z) = 1 4 (1 + 2z 1 + z 2) frequency response H(ˆω) = H ( z = e jˆω) = 1 2 magnitude, phase ( ) 1 + cos(ˆω) e jˆω 31 Fir Hanning
4 1.1 Transfer Function The z-transform transforms a discrete-time signal x[n] a sequence of numbers into the representation through a complexvalued function X( ) of the complex variable z, X(z). 1 For a given discrete-time signal x[n], its z-transform is defined by 2 x[n] X(z) = n= x[n]z n. To obtain the transfer function of the Hanning filter, we start with its specifying difference-equation and use the Ansatz x[n] X(z), y[n] Y (z), that is, we argue that, whatever are the signals input to and output from the filter, x[n] and y[n], respectively, there exist z-transform representations for them. 1 The complex variable z gives the name to the transformation. The z- transform is closely related to the characteristic function often used probability theory and statistics. 2 We use here the so-called bilateral z-transform that is most often applied in signal processing with its stationary problems; signals are always present, they extend from to +. There exists another z-transform, the unilateral z-transform, that is most often applied in control with its transient problems; signals start at zero and extend only to +. Problems treated in control are initial value problems. Note that there are some subtle differences in the properties of these two z-transforms; especially, the left-shift theorem involves in the unilateral z-transform the initial values of the signal or the system, whereas the bilateral z-transform does not. For a good and nevertheless short introduction to the bilateral z-transform you might want to study [MSY98, Chapter 7]; for a more in-depth treatment, discussing also the differences between unilateral and bilateral z-transform, you might consult [OWN97] or any newer edition of this text. 31 Fir Hanning
5 Next, we use the shift property of the z-transform stating that x[n] X(z) = x[n 1] z 1 X(z). Finally, using the linearity property of the z-transform, we may transform the given difference equation into the z domain: ( ) X(z) + 2z 1 X(z) + z 2 X(z) Y (z) = 1 4 = 1 4 ( 1 + 2z 1 + z 2) X(z). The transfer function is defined as the ratio of output signal to input signal in the z-domain description, H(z) ˆ= Y (z)/x(z); we thus obtain transfer function ˆ= H(z) ˆ= Y (z) X(z) = 1 ( 1 + 2z 1 + z 2) Frequency Response The frequency response of a (linear) system is the response of the system here the Hanning filter to a sinusoidal input signal, taken as a function of the frequency of the input sinusoid. Such a definition makes sense if the system is a linear system, which reacts to a sinusoidal input with a sinusoidal output of the same frequency; 3 what the linear system does to the input sinusoidal signal is that it changes its amplitude and its phase, and this amplitude- and phase change is described by the frequency response. Consider a discrete-time sinusoidal signal a sequence, that might be the input to the considered system, x[n] = cos ( nˆω 0 + φ ), 3 Note that the input sinusoidal signal is taken to exist over the complete time axis < n <. 31 Fir Hanning
6 where ˆω 0 is the discrete-time radian frequency, which is in the interval 0 ˆω 0 π, and where φ is its phase. Using Euler s formula, we can express this sinusoidal signal as x[n] = cos ( nˆω 0 + φ ), = 1 (e j(nˆω0+φ) + e j(nˆω0+φ)) 2 = 1 ( e jnˆω0 e jφ + e jnˆω0 e jφ), 2 that is, the sinusoidal signal is the sum of two basic complex exponential components. Because the considered systems are linear systems, for which superposition holds, it is sufficient to consider only complex exponentials as input signals the response to the initially considered sinusoidal signal can then be obtained by superposition of the responses to the complex exponential inputs. 4 Thus, the complex exponentials are the basic building blocks, and we continue to analyze what happens at the output of a linear system if we apply at the input signals of the form 5 x[n] = Ae jφ e jnˆω0, < n. (1) To compute the output of our Hanning filter if we apply at its input the complex exponential signal from (1), we just insert 4 The superposition principle holding for linear systems is the reason that the frequency response completely describes this kind of systems, that is, given the frequency response we may compute the output signal of the system for any given input signal: Due to the Fourier theory, any signal any practically relevant signal can be described by its Fourier series (or by its Fourier transform, if needed); we know for each component sinusoidal signal of this Fourier series its response at the output of the considered system; because the system is linear, a linear combination at the system s input the Fourier series leads to a corresponding linear combination at the system s output (this is the superposition principle). 5 We use now radian frequencies in the interval π ˆω 0 π. 31 Fir Hanning
7 this input signal x[n] into the difference equation describing the Hanning filter; we obtain y[n] = 1 ( ) x[n] + 2x[n 1] + x[n 2] 4 = 1 ) (Ae jφ e jnˆω0 + 2Ae jφ e j(n 1)ˆω0 + Ae jφ e j(n 2)ˆω0 4 = 1 ) (1 + 2e jˆω0 + e j2ˆω0 } 4 } Ae jφ {{ e jnˆω0 }. (2) {{} input x[n] ˆ= H(ˆω 0 ) The expression termed H(ˆω 0 ) in the above equation is independent of time n and is only dependent on the system to be described; it is the frequency response of the Hanning filter. We immediately see that H(ˆω 0 ) = H ( z = e jˆω0 ), that is, we obtain the frequency response H( ) by evaluating the transfer function H( ) on the unit circle. 6 From formula (2) we indeed see that the complex exponential signal x[n] = Ae jφ e jnˆω0 at the input to the linear filter re-appears at its output, but it is changed by the complex amplitude H(ˆω 0 ), which depends on the input frequency ˆω 0. By the mentioned superposition argument we see, in turn, that for the linear filter we indeed have sinus in = sinus with identical frequency out. To more easily see the amplitude- and phase change the frequency response H(ˆω) imposes to an input sinusoidal signal with a certain frequency ˆω, we next express H(ˆω) as the product of 6 The input variable z to the transfer function H( ) is a complex variable; therefore, its domain is the complex plane which we call the z plane. The values z = e jˆω for π ˆω π define a circle with center at the origin and with radius one in the z plane. 31 Fir Hanning
8 a real-valued function and a complex exponential function: H(ˆω) = 1 4 (1 + 2e jˆω + e j2ˆω) = 1 4 e jˆω( e +jˆω e jˆω) = 1 2 ( ) 1 + cos(ˆω) e jˆω = R(ˆω) e jˆω, where we have, to obtain the second but last equality, again used Euler s formula. 7 We now see that the real-valued function R(ˆω) contributes to the amplitude- as well as to the phase change that an input sinusoidal signal undergoes by passing the filter, 8 and that the complex exponential function only contributes to the phase change. Moreover, we see that this phase change is, for the considered Hanning filter, just ˆω, that is, the Hanning filter has a linear phase. We finally note that any Fir filter having a symmetric sequence of coefficients in its describing difference equation will have a linear phase as the Hanning filter has. With symmetric sequence of coefficients we mean that the first coefficient is equal to the last, the second coefficient is equal to the second but last, and so on. Likewise, any Fir filter having an anti-symmetric sequence of coefficients in its describing difference equation will also have such a linear phase. We mean with anti-symmetric sequence of coefficients that the first coefficient is equal to the negative of the last, the second coefficient is equal to the negative of the second but last, and so on. In 7 Here we have used Euler s formula in inverse direction, e jϕ +e jϕ = 2cos(ϕ). 8 Because the real-valued function obtained for the presently considered Hanning filter, R(ˆω) = (1 + cos(ˆω))/2, is always non-negative, it only contributes to the amplitude change. In general, R(ˆω) might become negative at a certain frequency, indicating there a phase jump of ±π. 31 Fir Hanning
9 the anti-symmetric case, the cos terms in R(ˆω) are replaced by sin terms; also note in this context that the imaginary unit j might be written as j = e jπ/2, an equality that might become useful if we sketch amplitude- and phase responses by hand. Magnitude- and Phase-Response 4 Magnitude of FIR Filter with Coefficients {1,2,1} 3 magnitude π 0 discrete radian frequency π π Phase of FIR Filter with Coefficients {1,2,1} phase 0 π π 0 discrete radian frequency π 31 Fir Hanning
10 1.3 Poles and Zeros The transfer function H(z) = (1 + 2z 1 + z 2 )/4 can be expressed in the following different forms: H(z) = 1 4 = 1 4 ( 1 + 2z 1 + z 2) (3) ( 1 + z 1)( 1 + z 1). (4) If we multiply by 1 = z 2 /z 2 we obtain two further equivalent forms H(z) = 1 4 = 1 4 z 2 + 2z + 1 z 2 (5) z + 1 z z + 1 z. (6) The factored form (6) clearly shows that the z-value z = 1 makes the transfer function zero: H(z = 1) 0. That value of z is called a zero of the transfer function; indeed, we even have a double zero at z = 1 because the factor z + 1 appears twice in (6). Equation (6) also shows that H(z) for z 0. Values of z for which the transfer function H(z) is undefined (infinite) are called the poles of H(z). In the present case we see that the term z 2 in the denominator of (5) represents two poles at z = 0, or, expressed differently, that H(z) has a double pole (a second-order pole) at z = 0. We see that the transfer function H(z) expressed in form (3) is a polynomial in z 1. Such a polynomial transfer-function form not just happens for the presently considered Hanning filter, but any Fir filter will have a polynomial transfer function. By 31 Fir Hanning
11 generalizing the above argumentation, we see than that any Fir filter will have only poles at the origin z = 0. Because poles of a filter (of a system) are responsible for its stability, and because a discrete-time filter is stable if and only if all its poles lie strictly inside of the unit circle in the z-plane, we see that any Fir filter is always stable. This finding is, of course, not a surprise, because the computations involved in computing a Fir filter s output sample only use samples of the input signal but no previously computed output samples, that is, there are only feed-forward computations but no feed-back computations. Pole-Zero Plot in the Complex z-plane The Complex z Plane (1 + 2z 1 + z 2) 1 H(z) = = 1 4 = 1 4 (1 + z 1)( 1 + z 1) z + 1 z z + 1 z Imaginary Part unit circle Real Part Note that the locations of both, the poles and the zeros, are also clear when H(z) is written in the form (4): Each factor of the form (1 az 1 ) in the presently considered situation a = 1 can always be expressed as (1 az 1 ) = (z a) z. 31 Fir Hanning
12 2 Signal Flow Graphs Recall the definition of our Hanning filter in the time domain, see page 1. Omitting here for simplicity the scaling factor 1/4, we have ( ) y[n] = x[n] + 2x[n 1] + x[n 2]. (7) We obviously see that to compute the output sample at time n, y[n], we need the following: First, a means for multiplying (possibly delayed) input-signal values x[ ] by the filter coefficients, which happen to be here {1, 2, 1}; second, a means for adding the scaled (multiplied) input-signal values; and third, a means for obtaining delayed versions of the input-signal samples. It is useful to represent the operations of (7) as a block diagram a so-called signal-flow graph or dynamic diagram. Such representations may lead to better insight about the properties of the system and about alternative ways to implement it. From the discussion above it is clear that the building blocks for dynamic diagrams for signal-flow graphs are multiplier, adder, and unit delay. Thereby, the multiplier is special in that one operand is constant the filter coefficient 9 and only the other operand changes the signal sample; the complete adder is usually realized by several two-input adders; and the unit delay is an element whose hardware implementation performs acquiring a sample value, storing it in memory for one clock cycle, and then releasing it to the output. Delays by more than one time unit are implemented (from a dynamic diagram point 9 In the presently considered Fir filters, the coefficients are constants; in the later considered adaptive filters, which often base on Fir filter structures, these coefficients are no longer constants (which are, in the simple Fir filter case, once determined at the design-phase of the filter), but, instead, are coefficients that change adapt during system operation. 31 Fir Hanning
13 of view) by a cascade of several unit delays. An M-unit delay thus requires M memory cells configured as a shift register; in a computer memory, such a shift register may be implemented as a circular buffer Direct Form The most simple block diagram of the Hanning filter (7) is the so-called direct form. It computes the output sample y[n] by ( ) y[n] = x[n 2] + 2x[n 1] + x[n]. Direct-Form Block Diagram of Hanning Filter x[n] z 1 x[n 1] z 1 x[n 2] y[n] 10 In a shift register, the register contents the data are moved (shifted); in a circular buffer, the register contents remain in place, but the pointer addressing them is moved. 31 Fir Hanning
14 2.2 Transposed Form Obviously, we may compute differently as compared to the direct form. For example, computing as ( ) y[n] = x[n] + 2x[n 1] + x[n 2] leads to the transposed form. Indeed, using the internal signal names in the block diagram below, we have y[n] = x[n] + v 1 [n 1], v 1 [n] = 2x[n] + v 2 [n 1] = x[n] + 2x[n 1] + v 2 [n 2], v 2 [n] = x[n] = x[n] + 2x[n 1] + x[n 2]. Transposed-Form Block Diagram of Hanning Filter x[n] 2 z 1 + z 1 + v 2 [n] v 1 [n] y[n] 31 Fir Hanning
15 2.3 Cascade Form Still another form of block diagram might be obtained if we start from the factored transfer function in (4); again omitting the scaling factor 1/4, we have the cascade of two systems with transfer function ( 1 + z 1) each. Because ( 1 + z 1) corresponds to the difference equation y[n] = x[n] + x[n 1] for an input sequence x[n] and corresponding output sequence y[n], we have for the complete Hanning filter the block diagram below. Cascade-Form Block Diagram of Hanning Filter x[n] z 1 x[n 1] + y 1[n] z 1 y 1 [n 1] + y[n] 31 Fir Hanning
16 We finally note that in general many block diagrams implement the same Fir filter in the sense that the externally seen behavior from input to output will be the same; the discussed direct form, transposed form, and cascade form are just examples. Different block diagrams represent different internal computations or different orders of computation. Therefore, different block diagrams that implement the same input-output characteristic may have very different characteristics concerning there internal behavior. Especially, finite word-length effects are important if the filter is implemented in fixed-point arithmetic, where round-off noise and over- and underflow are important real-world problems which depend on the internal order of computation. References [MSY98] James H. McClellan, Ronald W. Schafer, and Mark A. Yoder. Dsp First: A Multimedia Approach. Prentice-Hall Inc., [OWN97] Alan V. Oppenheim, Alan S. Willsky, and S. Hamid Nawab. Signals and Systems. Prentice-Hall Inc., Englewood Cliffs, N.J., 2nd edition, Fir Hanning
An Iir-Filter Example: A Butterworth Filter
An Iir-Filter Example: A Butterworth Filter Josef Goette Bern University of Applied Sciences, Biel Institute of Human Centered Engineering - microlab JosefGoette@bfhch February 7, 2017 Contents 1 Introduction
More informationLecture 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 informationDSP Configurations. responded with: thus the system function for this filter would be
DSP Configurations In this lecture we discuss the different physical (or software) configurations that can be used to actually realize or implement DSP functions. Recall that the general form of a DSP
More informationCh. 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 informationDiscrete-Time Systems
FIR Filters With this chapter we turn to systems as opposed to signals. The systems discussed in this chapter are finite impulse response (FIR) digital filters. The term digital filter arises because these
More informationTransform 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 informationECE4270 Fundamentals of DSP Lecture 20. Fixed-Point Arithmetic in FIR and IIR Filters (part I) Overview of Lecture. Overflow. FIR Digital Filter
ECE4270 Fundamentals of DSP Lecture 20 Fixed-Point Arithmetic in FIR and IIR Filters (part I) School of ECE Center for Signal and Information Processing Georgia Institute of Technology Overview of Lecture
More informationFrom Wikipedia, the free encyclopedia
Z-transform - Wikipedia, the free encyclopedia Z-transform Page 1 of 7 From Wikipedia, the free encyclopedia In mathematics and signal processing, the Z-transform converts a discrete time domain signal,
More informationDetailed Solutions to Exercises
Detailed Solutions to Exercises Digital Signal Processing Mikael Swartling Nedelko Grbic rev. 205 Department of Electrical and Information Technology Lund University Detailed solution to problem E3.4 A
More informationSignals 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 informationHow 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 informationLECTURE NOTES DIGITAL SIGNAL PROCESSING III B.TECH II SEMESTER (JNTUK R 13)
LECTURE NOTES ON DIGITAL SIGNAL PROCESSING III B.TECH II SEMESTER (JNTUK R 13) FACULTY : B.V.S.RENUKA DEVI (Asst.Prof) / Dr. K. SRINIVASA RAO (Assoc. Prof) DEPARTMENT OF ELECTRONICS AND COMMUNICATIONS
More informationA.1 THE SAMPLED TIME DOMAIN AND THE Z TRANSFORM. 0 δ(t)dt = 1, (A.1) δ(t)dt =
APPENDIX A THE Z TRANSFORM One of the most useful techniques in engineering or scientific analysis is transforming a problem from the time domain to the frequency domain ( 3). Using a Fourier or Laplace
More informationVU Signal and Image Processing
052600 VU Signal and Image Processing Torsten Möller + Hrvoje Bogunović + Raphael Sahann torsten.moeller@univie.ac.at hrvoje.bogunovic@meduniwien.ac.at raphael.sahann@univie.ac.at vda.cs.univie.ac.at/teaching/sip/18s/
More informationDISCRETE-TIME SIGNAL PROCESSING
THIRD EDITION DISCRETE-TIME SIGNAL PROCESSING ALAN V. OPPENHEIM MASSACHUSETTS INSTITUTE OF TECHNOLOGY RONALD W. SCHÄFER HEWLETT-PACKARD LABORATORIES Upper Saddle River Boston Columbus San Francisco New
More informationUNIT - III PART A. 2. Mention any two techniques for digitizing the transfer function of an analog filter?
UNIT - III PART A. Mention the important features of the IIR filters? i) The physically realizable IIR filters does not have linear phase. ii) The IIR filter specification includes the desired characteristics
More informationz-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 informationUNIT V FINITE WORD LENGTH EFFECTS IN DIGITAL FILTERS PART A 1. Define 1 s complement form? In 1,s complement form the positive number is represented as in the sign magnitude form. To obtain the negative
More informationDiscrete-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 informationLet 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 informationLecture 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 informationLecture 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 informationZ-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 informationSignals, Instruments, and Systems W5. Introduction to Signal Processing Sampling, Reconstruction, and Filters
Signals, Instruments, and Systems W5 Introduction to Signal Processing Sampling, Reconstruction, and Filters Acknowledgments Recapitulation of Key Concepts from the Last Lecture Dirac delta function (
More informationDiscrete-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 informationReview 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 informationELEG 305: Digital Signal Processing
ELEG 305: Digital Signal Processing Lecture 19: Lattice Filters Kenneth E. Barner Department of Electrical and Computer Engineering University of Delaware Fall 2008 K. E. Barner (Univ. of Delaware) ELEG
More informationEE 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 informationEE102B Signal Processing and Linear Systems II. Solutions to Problem Set Nine Spring Quarter
EE02B Signal Processing and Linear Systems II Solutions to Problem Set Nine 202-203 Spring Quarter Problem 9. (25 points) (a) 0.5( + 4z + 6z 2 + 4z 3 + z 4 ) + 0.2z 0.4z 2 + 0.8z 3 x[n] 0.5 y[n] -0.2 Z
More information2. Typical Discrete-Time Systems All-Pass Systems (5.5) 2.2. Minimum-Phase Systems (5.6) 2.3. Generalized Linear-Phase Systems (5.
. Typical Discrete-Time Systems.1. All-Pass Systems (5.5).. Minimum-Phase Systems (5.6).3. Generalized Linear-Phase Systems (5.7) .1. All-Pass Systems An all-pass system is defined as a system which has
More informationNeed for transformation?
Z-TRANSFORM In today s class Z-transform Unilateral Z-transform Bilateral Z-transform Region of Convergence Inverse Z-transform Power Series method Partial Fraction method Solution of difference equations
More informationDSP Design Lecture 2. Fredrik Edman.
DSP Design Lecture Number representation, scaling, quantization and round-off Noise Fredrik Edman fredrik.edman@eit.lth.se Representation of Numbers Numbers is a way to use symbols to describe and model
More informationUNIVERSITY 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 informationDiscrete-Time Signals: Time-Domain Representation
Discrete-Time Signals: Time-Domain Representation 1 Signals represented as sequences of numbers, called samples Sample value of a typical signal or sequence denoted as x[n] with n being an integer in the
More informationA 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 informationTheory and Problems of Signals and Systems
SCHAUM'S OUTLINES OF Theory and Problems of Signals and Systems HWEI P. HSU is Professor of Electrical Engineering at Fairleigh Dickinson University. He received his B.S. from National Taiwan University
More informationy[n] = = h[k]x[n k] h[k]z n k k= 0 h[k]z k ) = H(z)z n h[k]z h (7.1)
7. The Z-transform 7. Definition of the Z-transform We saw earlier that complex exponential of the from {e jwn } is an eigen function of for a LTI System. We can generalize this for signals of the form
More informationDiscrete-Time Signals: Time-Domain Representation
Discrete-Time Signals: Time-Domain Representation 1 Signals represented as sequences of numbers, called samples Sample value of a typical signal or sequence denoted as x[n] with n being an integer in the
More informationZ 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 informationDiscrete 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 informationIT 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 informationTransform Representation of Signals
C H A P T E R 3 Transform Representation of Signals and LTI Systems As you have seen in your prior studies of signals and systems, and as emphasized in the review in Chapter 2, transforms play a central
More informationE : 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 informationSIGNALS AND SYSTEMS. Unit IV. Analysis of DT signals
SIGNALS AND SYSTEMS Unit IV Analysis of DT signals Contents: 4.1 Discrete Time Fourier Transform 4.2 Discrete Fourier Transform 4.3 Z Transform 4.4 Properties of Z Transform 4.5 Relationship between Z
More informationOn the Frequency-Domain Properties of Savitzky-Golay Filters
On the Frequency-Domain Properties of Savitzky-Golay Filters Ronald W Schafer HP Laboratories HPL-2-9 Keyword(s): Savitzky-Golay filter, least-squares polynomial approximation, smoothing Abstract: This
More informationVALLIAMMAI ENGINEERING COLLEGE. SRM Nagar, Kattankulathur DEPARTMENT OF INFORMATION TECHNOLOGY. Academic Year
VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur- 603 203 DEPARTMENT OF INFORMATION TECHNOLOGY Academic Year 2016-2017 QUESTION BANK-ODD SEMESTER NAME OF THE SUBJECT SUBJECT CODE SEMESTER YEAR
More informationDiscrete-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 informationThe Laplace Transform
The Laplace Transform Syllabus ECE 316, Spring 2015 Final Grades Homework (6 problems per week): 25% Exams (midterm and final): 50% (25:25) Random Quiz: 25% Textbook M. Roberts, Signals and Systems, 2nd
More informationSignals and Systems: Introduction
Dependent variable Signals and Systems: Introduction What is a signal? Signals may describe a wide variety of physical phenomena. The information in a signal is contained in a pattern of variations of
More informationAnalysis of Finite Wordlength Effects
Analysis of Finite Wordlength Effects Ideally, the system parameters along with the signal variables have infinite precision taing any value between and In practice, they can tae only discrete values within
More informationLecture 5 - Assembly Programming(II), Intro to Digital Filters
GoBack Lecture 5 - Assembly Programming(II), Intro to Digital Filters James Barnes (James.Barnes@colostate.edu) Spring 2009 Colorado State University Dept of Electrical and Computer Engineering ECE423
More informationLecture 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 informationRecursive Gaussian filters
CWP-546 Recursive Gaussian filters Dave Hale Center for Wave Phenomena, Colorado School of Mines, Golden CO 80401, USA ABSTRACT Gaussian or Gaussian derivative filtering is in several ways optimal for
More information6.003: Signals and Systems
6.003: Signals and Systems Z Transform September 22, 2011 1 2 Concept Map: Discrete-Time Systems Multiple representations of DT systems. Delay R Block Diagram System Functional X + + Y Y Delay Delay X
More informationDiscrete-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 informationDigital 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 informationGeneralizing the DTFT!
The Transform Generaliing the DTFT! The forward DTFT is defined by X e jω ( ) = x n e jωn in which n= Ω is discrete-time radian frequency, a real variable. The quantity e jωn is then a complex sinusoid
More informationComments on Hilbert Transform Based Signal Analysis
Brigham Young University Department of Electrical and Computer Engineering 459 Clyde Building Provo, Utah 84602 Comments on Hilbert Transform Based Signal Analysis David G. Long, Ph.D. Original: 15 June
More informationUse: 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# 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 informationVery 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 informationLinear 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 information1 Introduction & Objective. 2 Warm-up. Lab P-16: PeZ - The z, n, and O! Domains
DSP First, 2e Signal Processing First Lab P-6: PeZ - The z, n, and O! Domains The lab report/verification will be done by filling in the last page of this handout which addresses a list of observations
More informationTopic 3: Fourier Series (FS)
ELEC264: Signals And Systems Topic 3: Fourier Series (FS) o o o o Introduction to frequency analysis of signals CT FS Fourier series of CT periodic signals Signal Symmetry and CT Fourier Series Properties
More informationThe Laplace Transform
The Laplace Transform Generalizing the Fourier Transform The CTFT expresses a time-domain signal as a linear combination of complex sinusoids of the form e jωt. In the generalization of the CTFT to the
More informationDigital 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 informationDigital 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 informationDiscrete 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 informationCMPT 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 information3 What You Should Know About Complex Numbers
3 What You Should Know About Complex Numbers Life is complex it has a real part, and an imaginary part Andrew Koenig. Complex numbers are an extension of the more familiar world of real numbers that make
More informationDigital Filters. Linearity and Time Invariance. Linear Time-Invariant (LTI) Filters: CMPT 889: Lecture 5 Filters
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
More informationMultimedia 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 informationELEG 305: Digital Signal Processing
ELEG 305: Digital Signal Processing Lecture 4: Inverse z Transforms & z Domain Analysis Kenneth E. Barner Department of Electrical and Computer Engineering University of Delaware Fall 008 K. E. Barner
More informationECE-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 informationDigital Filter Structures. Basic IIR Digital Filter Structures. of an LTI digital filter is given by the convolution sum or, by the linear constant
Digital Filter Chapter 8 Digital Filter Block Diagram Representation Equivalent Basic FIR Digital Filter Basic IIR Digital Filter. Block Diagram Representation In the time domain, the input-output relations
More informationDigital Filter Implementation 1
Digital Filter Implementation 1 V 5, November 13, 2017 Christian Feldbauer, Bernhard Geiger, Josef Kulmer, {kulmer}@tugraz.at Signal Processing and Speech Communication Laboratory, www.spsc.tugraz.at Inffeldgasse
More informationTwo-Dimensional Systems and Z-Transforms
CHAPTER 3 Two-Dimensional Systems and Z-Transforms In this chapter we look at the -D Z-transform. It is a generalization of the -D Z-transform used in the analysis and synthesis of -D linear constant coefficient
More informationLECTURE NOTES IN AUDIO ANALYSIS: PITCH ESTIMATION FOR DUMMIES
LECTURE NOTES IN AUDIO ANALYSIS: PITCH ESTIMATION FOR DUMMIES Abstract March, 3 Mads Græsbøll Christensen Audio Analysis Lab, AD:MT Aalborg University This document contains a brief introduction to pitch
More informationNotice the minus sign on the adder: it indicates that the lower input is subtracted rather than added.
6.003 Homework Due at the beginning of recitation on Wednesday, February 17, 010. Problems 1. Black s Equation Consider the general form of a feedback problem: + F G Notice the minus sign on the adder:
More informationLecture 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 informationAlgebra III/Trigonometry CP
East Penn School District Secondary Curriculum A Planned Course Statement for Algebra III/Trigonometry CP Course # 330 Grade(s) 10,11,12 Department: Mathematics ength of Period (mins.) 42 Total Clock Hours:
More informationChap 2. Discrete-Time Signals and Systems
Digital Signal Processing Chap 2. Discrete-Time Signals and Systems Chang-Su Kim Discrete-Time Signals CT Signal DT Signal Representation 0 4 1 1 1 2 3 Functional representation 1, n 1,3 x[ n] 4, n 2 0,
More informationModule 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 informationUNIT 1. SIGNALS AND SYSTEM
Page no: 1 UNIT 1. SIGNALS AND SYSTEM INTRODUCTION A SIGNAL is defined as any physical quantity that changes with time, distance, speed, position, pressure, temperature or some other quantity. A SIGNAL
More informationThe Laplace Transform
The Laplace Transform Introduction There are two common approaches to the developing and understanding the Laplace transform It can be viewed as a generalization of the CTFT to include some signals with
More informationDHANALAKSHMI 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 informationLECTURE 12 Sections Introduction to the Fourier series of periodic signals
Signals and Systems I Wednesday, February 11, 29 LECURE 12 Sections 3.1-3.3 Introduction to the Fourier series of periodic signals Chapter 3: Fourier Series of periodic signals 3. Introduction 3.1 Historical
More informationDigital 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 informationZ-Transform. The Z-transform is the Discrete-Time counterpart of the Laplace Transform. Laplace : G(s) = g(t)e st dt. Z : G(z) =
Z-Transform The Z-transform is the Discrete-Time counterpart of the Laplace Transform. Laplace : G(s) = Z : G(z) = It is Used in Digital Signal Processing n= g(t)e st dt g[n]z n Used to Define Frequency
More informationSignal Processing First Lab 11: PeZ - The z, n, and ˆω Domains
Signal Processing First Lab : PeZ - The z, n, and ˆω Domains The lab report/verification will be done by filling in the last page of this handout which addresses a list of observations to be made when
More informationUniversity of Illinois at Urbana-Champaign ECE 310: Digital Signal Processing
University of Illinois at Urbana-Champaign ECE 0: Digital Signal Processing Chandra Radhakrishnan PROBLEM SET : SOLUTIONS Peter Kairouz Problem. Hz z 7 z +/9, causal ROC z > contains the unit circle BIBO
More informationLike 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 informationChapter 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 informationEE 3054: Signals, Systems, and Transforms Spring A causal discrete-time LTI system is described by the equation. y(n) = 1 4.
EE : Signals, Systems, and Transforms Spring 7. A causal discrete-time LTI system is described by the equation Test y(n) = X x(n k) k= No notes, closed book. Show your work. Simplify your answers.. A discrete-time
More informationZ-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 informationDigital Filter Structures
Chapter 8 Digital Filter Structures 清大電機系林嘉文 cwlin@ee.nthu.edu.tw 03-5731152 8-1 Block Diagram Representation The convolution sum description of an LTI discrete-time system can, in principle, be used to
More information8. z-domain Analysis of Discrete-Time Signals and Systems
8. z-domain Analysis of Discrete-Time Signals and Systems 8.. Definition of z-transform (0.0-0.3) 8.2. Properties of z-transform (0.5) 8.3. System Function (0.7) 8.4. Classification of a Linear Time-Invariant
More informationDiscrete-Time Signals and Systems. The z-transform and Its Application. The Direct z-transform. Region of Convergence. Reference: Sections
Discrete-Time Signals and Systems The z-transform and Its Application Dr. Deepa Kundur University of Toronto Reference: Sections 3. - 3.4 of John G. Proakis and Dimitris G. Manolakis, Digital Signal Processing:
More informationProblem Value
GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM DATE: 30-Apr-04 COURSE: ECE-2025 NAME: GT #: LAST, FIRST Recitation Section: Circle the date & time when your Recitation
More informationButterworth Filter Properties
OpenStax-CNX module: m693 Butterworth Filter Properties C. Sidney Burrus This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 3. This section develops the properties
More information