Analog vs. discrete signals

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1 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 are defined only at discrete times, that is, at a discrete set of values of an independent variable. Frequently, discrete signals arise from sampling their continuous counterparts. The result is a sequence of numbers defined by s[n] = s(t) = s(nt ), t=nt where n Z and T is a sampling period. The quantity F s = 1/T is known as sampling frequency. A discrete-time signal s[n] whose amplitude takes values from a finite set of K numbers {a k } K k=1 is known as a digital signal. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 1/32

2 Different signal formats Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 2/32

3 Continuous-time systems A continuous-time system (CTS) is a system which transforms a continuous-time input signal x(t) into a continuous-time output signal y(t). For example: y(t) = t x(τ)dτ. Symbolically, the input-output relation of a continuous-time system can be represented by H : x(t) y(t) or H{x(t)} = y(t). In real-life, CTS are implemented using analog electronic circuits. The physical implementation of a CTS is called an analog system. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 3/32

4 Discrete-time systems A system that transforms a discrete-time input signal x[n] into a discrete-time output signal y[n], is called a discrete-time system. Symbolically, the input-output relation of a discrete-time system is represented by H : x[n] y[n] or H{x[n]} = y[n]. The physical implementation of discrete-time systems can be done either in software or hardware. In both cases, the underlying physical systems consist of digital electronic circuits designed to manipulate logical information or physical quantities represented in digital form by binary electronic signals. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 4/32

5 Analog-to-digital conversion Quantization converts a continuous-amplitude signal x(t) into a discrete-amplitude signal x d [n]. In theory, we are dealing with discrete-time signals; in practice, we are dealing with digital signals. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 5/32

6 Digital-to-analog conversion The conversion of a discrete-time signal into continuous time form is done using a digital-to-analog (D/A) converter. An ideal D/A converter or interpolator essentially fills the gaps between the samples of a sequence of numbers to create a continuous-time function. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 6/32

7 Analog, digital, and mixed signal processing Typical implementation: Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 7/32

8 Applications of DSP Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 8/32

Discrete-time signals We will use the following notation: x[n] to represent the n-th sample of the sequence; {x[n]} N 2 n=n 1 to represent the samples in the range N 1 n N 2; {x[n]} to

9 Discrete-time signals We will use the following notation: x[n] to represent the n-th sample of the sequence; {x[n]} N 2 n=n 1 to represent the samples in the range N 1 n N 2; {x[n]} to represent the entire sequence. There is a number of different ways to represent discrete signals: Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 9/32

10 Discrete-time signals (continued) The energy of a sequence x[n] is defined as E x := [ x[n] 2 = sum(x.*conj(x)) = Ex n ]. The power of a sequence x[n] is defined as the average energy per sample P x := lim L { 1 2L + 1 L n= L x[n] 2 } [ = Ex/length(x) = Px When signal x[n] represents a physical signal, both quantities are directly related to the energy and power of the signal. Note that when the duration of {x[n]} is finite, its E x is always finite, while P x is equal to zero (assuming that the signal values outside of its support are equal to zero). ]. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 10/32

11 Some elementary signals Unit sample sequence (aka discrete Dirac ) { 1, n = 0 δ[n] =. 0, n 0 Unit step sequence (aka discrete Heavyside ) { 1, n 0 n u[n] = = δ[k]. 0, n < 0 Sinusoidal sequence x[n] = A cos(ω 0 n + ϕ), Exponential sequence k= x[n] = A a n, A, a C. Complex exponential sequence with [ω 0 ] = radians sample. x[n] = A a jω0n = A cos(ω 0 n) + ja sin(ω 0 n) Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 11/32

12 Periodic sequence (continued) A sequence x[n] is called periodic if x[n] = x[n + N], n Z. The smallest N for which it holds is called the fundamental period of {x[n]}. A sinusoidal sequence is periodic if cos(ω 0 n + ϕ) = cos(ω 0 (n + N) + ϕ) = cos(ω 0 n + ω 0 N + ϕ), which necessitates ω 0 N = 2πk, k Z. Thus, for a given N, there are exactly N distinct sinusoidal sequences with frequencies ω k = 2πk, with k = 0, 1,..., N 1. N Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 12/32

13 Discrete-time systems Causality H is called causal if, for any n 0, y[n 0 ] is determined by the values of x[n] for n n 0 only. y[n] = 1/3 (x[n] + x[n 1] + x[n 2]) y[n] = median (x[n 1], x[n], x[n + 1]) (causal) (non-causal) BIBO stability H is said to be stable in the Bounded-Input Bounded-Output (BIBO) sense, if x[n] M x < implies y[n] M y < for some positive constants M x and M y. y[n] = (1/L) L 1 k=0 x[n k] (stable) y[n] = n k= x[k] (not stable) Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 13/32

14 Discrete-time systems (continued) Linearity H is called linear if for every a 1 and a 2 (either real or complex) and every input signals x 1 [n] and x 2 [n] it holds that H {a 1 x 1 [n] + a 2 x 2 [n]} = a 1 H{x 1 } + a 2 H{x 2 } y[n] = x[n] x[n 1] y[n] = (x[n]) 2 (linear) (non-linear) Time-invariance (TI) H is called time-invariant if and only if y[n] = H{x[n]} = y[n n 0 ] = H{x[n n 0 ]} y[n] = cos(x[n]) y[n] = n x[n]w[n], for some fixed {w[n]} (TI) (not TI) Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 14/32

15 Block diagrams and signal flow graphs Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 15/32

16 Signal flow graph (example) In the above example, we have w[n] = x[n] + a w[n 1] y[n] = w[n] + b w[n 1] (input node) (output node) After a simple manipulation to eliminate w[n], we obtain y[n] = x[n] + b x[n 1] + a y[n 1]. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 16/32

17 Folding and time-shifting The operations of shifting and folding are not commutative. Indeed: x[n] shift x[n n 0 ] fold x[ n n 0 ] x[n] fold x[ n] shift x[ n+n 0 ] In MATLAB, discrete sequences can be plotted using stem(n, x, 'fill') xlabel('n') ylabel('x[n]') Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 17/32

18 Signal decomposition into impulses A time-shifted version δ[n k] of δ[n] is defined as { 1, n = k δ[n k] = 0, n k Using this sequence, one can redefine {x[n]} as given by For example: x[n] = k= x[k]δ[n k] Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 18/32

19 Impulse response Impulse response The impulse response h[n] of a system y[n] = H{x[n]} is defined as h[n] = H{δ[n]} Now, if H is linear time-invariant (LTI), then H{δ[n k]} = h[n k] and, consequently, { } H{x[n]} = H x[k]δ[n k] = x[k]h{δ[n k]} = x[k]h[n k] k k k Discrete convolution Thus, the response of an LTI system can be described as y[n] = H{x[n]} = x[k]h[n k] = x[n] h[n] k= Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 19/32

Understanding the discrete convolution Let us consider a case with x[n] = {1 2 3 4 5} and h[n] = { 1 2 1}. Our task is to compute y[3] = k x[k]h[3 k].

20 Understanding the discrete convolution Let us consider a case with x[n] = { } and h[n] = { 1 2 1}. Our task is to compute y[3] = k x[k]h[3 k]. Above, both {x[n]} and {h[n]} were converted to infinite-length sequences by means of zero-padding. This type of convolution is called linear. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 20/32

21 Numerical computation of convolution Suppose we want to convolve {h[n]} M 1 n=0 and N = 6. In this case, we have and {x[n]}n 1 n=0, with M = 3 One can see that: y[n] = 0 for n < 0 y[n] = 0 for n N + M 1 Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 21/32

22 Numerical computation of convolution (cont.) The operation of convolution can be expressed in the form of vector-matrix multiplication. Specifically, y[0] x[0] 0 0 x[0] y[1] x[1] x[0] 0 x[1] y[2] x[2] x[1] x[0] y[3] x[3] x[2] x[1] = h[0] x[2] h[1] x[3] = h[0] +...+h[2] y[4] x[4] x[3] x[2] x[4] h[2] y[5] x[5] x[4] x[3] x[5] y[6] 0 x[5] x[4] 0 y[7] 0 0 x[5] 0 } {{ } X 0 0 x[0] x[1] x[2] x[3] x[4] x[5] The convolution matrix X is a Toeplitz matrix (can be built using the m-function toeplitz). The m-function conv can be used to compute linear convolution in the same range 0 n N 1 (using conv(x,h, same )) in the full overlap range (using conv(x,h, valid )) in the entire range (using conv(x,h)) Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 22/32

23 Properties of LTI systems Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 23/32

24 Causality and stability Causality An LTI system is causal if h[n] = 0 for n < 0. Indeed, y[n] = + h[ 1]x[n + 1] + h[0]x[n] + h[1]x[n 1] +. So, y[n] won t depend on the future values x[n + 1], x[x + 2], x[x + 3],..., if h[n] = 0 for n < 0. Stability An LTI system is BIBO stable if and only if its impulse response is absolutely summable, that is, if n h[n] <. Indeed, we have y[n] = h[k]x[n k] k k h[k] x[n k] M x h[k]. k Thus, if S h = n h[n] < and M y = S h M x, then y[n] M y. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 24/32

25 Stability (example) Is an LTI system with h[n] = b a n u[n] BIBO stable? S h = n h[n] = b a n n=0 Recall that the sum of a geometric series {1 r r 2 r 3...} is given by n=0 Therefore, if a < 1, then S h = b r n = 1, if r < 1 1 r a n = b 1 a < n=0 Thus, the system if stable only if a < 1. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 25/32

26 Recursive implementation of LTI systems An LTI system H with h[n] = b a n u[n] is called an infinite impulse response (IIR) filter. In this case, y[n] = h[k]x[n k] = b x[n] + b a x[n 1] + b a 2 x[n 2] +... = k=0 = b x[n] + a (b x[n 1] + b a x[n 2] +...) = b x[n] + a y[n 1]. In this case, we said that y[n] is computed recursively. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 26/32

27 Linear constant coefficients difference equations In general, LTI systems can be described by linear constant-coefficient difference equations (LCCDE) of the following form. General recursive systems The IO relation of a general LTI system is given by N M y[n] = a k y[n k] + b k x[n k] k=1 where {a k } N k=1 and {b k} M k=1 are feedback and feedforward coefficients, respectively. k=0 N is known as the order of the system. If N = 0, then y[n] = (x h)[n], with h[n] = b n, n = 0, 1,..., M. In the above case, the system is causal and said to have a finite impulse response (FIR). Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 27/32

28 Real-time implementation of FIR filters In most real-time applications, convolution is computed using stream processing. The delay, τ < T, between the arrival of an input sample and the generation of the associated output sample is known as latency. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 28/32

29 Recursive FIR systems The moving average filter y[n] = 1 M + 1 M x[n k] k=0 is an FIR system with impulse response { (M + 1) 1, 0 n M h[n] =. 0, otherwise The system can be implemented as y[n] = y[n 1] + 1 (x[n] x[n M 1]), M + 1 which leads to a recursive implementation. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 29/32

30 Echo generation and reverberation The sound may consist of several components: direct sound, early reflections, and reverberations. Each components requires appropriate modelling. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 30/32

31 Echo generation and reverberation (cont.) A single echo can be modelled using an FIR filter of the form y[n] = x[n] + a x[n D]. A multiple echo can be modelled as an IIR system given by y[n] = x[n] + a x[n D] + a 2 x[n 2D] + a 3 x[n 3D] +... = = a y[n D] + x[n]. Thus, the problem of dereverberation amounts to finding an inverse filter to cancel the effect of the direct (model) filter. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 31/32

32 Main questions Given a system described by an LCCDE of the form we wish to be able to: N M y[n] = a k y[n k] + b k x[n k] k=1 1 Prove that the system is LTI. 2 Analytically determine its h[n]. k=0 3 Given an analytical expression for the input x[n], find an analytical expression for the output y[n]. 4 Given {a k, b k }, determine whether or not the system is stable. Z-transform All the above questions can be conveniently addressed using the tool of z-transform. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 1 32/32

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: