Digital Signal Processing Module 1 Analysis of Discrete time Linear Time - Invariant Systems Objective: 1. To understand the representation of Discrete time signals 2. To analyze the causality and stability concepts of Linear Shift Invariant (LSI) systems Introduction: Digital signals are discrete in both time (the independent variable) and amplitude (the dependent variable). Signals that are discrete in time but continuous in amplitude are referred to as discrete-time signals. A discrete-time system is one that processes a discrete-time input sequence to produce a discrete-time output sequence. There are many different kinds of such systems. One of the important kinds is Linear Shift Invariant (LSI) systems. Description: Discrete time Signals and Sequences Discrete-time signals are data sequences. A sequence of data is denoted {x[n]} or simply x[n] when the meaning is clear. The elements of the sequence are called samples. The index n associated with each sample is an integer. If appropriate, the range of n will be specified. Quite often, we are interested in identifying the sample where n = 0. This is done by putting an arrow under that sample. For instance, x n =,0.35,1,1.5, 0.6, 2, The arrow is often omitted if it is clear from the context which sample is x[0]. Sample values can either be real or complex. The terms discrete-time signals and sequences are used interchangeably. The time interval between samples is can be assumed to be normalized to 1 unit of time. So the corresponding normalized sampling frequency is 1Hz. If the actual sampling interval is T seconds, then the sampling frequency is given by f s = 1 T Some Elementary Sequences i) Unit Impulse Sequence The unit impulse sequence is defined by This is depicted graphically in Figure 1.1. Note that while the continuous-time unit impulse function is a mathematical object that cannot be physically realized, the unit impulse sequence can easily be generated.
Figure 1.1: The Unit Impulse Sequence ii) Unit Step Sequence The unit step sequence is one that has amplitude of zero for negative indices and amplitude of one for non-negative indices. u n = 0 n < 0 1 n 0 It is shown in Figure 1.2. Figure 1.2: The Unit Step Sequence iii) Exponential sequences The general form is If A and α are real numbers then the sequence is real. If 0 < α < 1 and A is positive, then the sequence values are positive and decrease with increasing n:
iv) Sinusoidal Sequences A sinusoidal sequence has the form This function can also be decomposed into its in-phase x i [q] and quadrature x q [n] components. This is a common practice in communications signal processing. It is shown in Figure 1.3. v) Complex Exponential Sequences Figure 1.3: The Sinusoidal Sequence Complex exponential sequences are essentially complex sinusoids. vi) Random Sequences The sample values of a random sequence are randomly drawn from a certain probability distribution. They are also called stochastic sequences. The two most common distributions are the Gaussian (normal) distribution and the uniform distribution. The zero-mean Gaussian distribution is often used to model noise. Figure 1.4 and Figure 1.5 show examples of uniformly distributed and Gaussian distributed random sequences respectively.
Figure 1.4: Uniformly distributed random sequence with amplitudes between -0.5 and 0.5. Types of Sequences Figure 1.5: Gaussian distributed random sequence with zero mean and unit variance. The discrete-time signals that we encounter can be classified in a number of ways. Some basic classifications that are of interest to us are described below. i) Real vs. Complex Signals A sequence is considered complex at least one sample is complex-valued. ii) iii) Finite vs. Infinite Length Signals Finite length sequences are defined only for a range of indices, say N1 ton2. The length of this finite length sequence is given by N2- N1 + 1. Causal Signals A sequence x[n] is a causal sequence if x[n] = 0 for n < 0.
iv) Symmetric Signals First consider a real-valued sequence {x[n]}. Even symmetry implies that x[n] = x[-n] and Odd symmetry implies that x[n] = -x[n] for all n. Any real-valued sequence can be decomposed into odd and even parts so that where the even part is given by and the odd part is given by A complex-valued sequence is conjugate symmetric if x[n] = x*[-n]. The sequence has conjugate anti-symmetry if x[n] = -x*[-n]. Analogous to real-valued sequences, any complexvalued sequence can be decomposed into its conjugate symmetric and conjugate antisymmetric parts: v) Periodic Signals A discrete-time sequence is periodic with a period of N samples if for all integer values of k. Note that N has to be a positive integer. If a sequence is not periodic, it is aperiodic or non-periodic. Consider a discrete-time sequence x [n] based on a sinusoid with angular frequency ω o : If this sequence is periodic with a period of N samples, then the following must be true: However, the left hand side can be expressed as and the cosine function is periodic with a period of 2π and therefore the right hand side of the above equation is given by for integer values of r. Comparing the above two equations, we have
where ω o = 2πf o. Since both r and N are integers, a discrete-time sinusoidal sequence is periodic if its frequency is a rational number. Otherwise, it is non-periodic. vi) Energy and Power Signals The energy of a finite length sequence x[n] is defined as while that for an infinite sequence is Note that the energy of an infinite length sequence may not be infinite. A signal with finite energy is usually referred to as an energy signal. The average power of a periodic sequence with a period of N samples is defined as and for non-periodic sequences, it is defined in terms of the following limit if it exists: A signal with finite average power is called a power signal. vii) Bounded Signals A sequence is bounded if every sample of the sequence has a magnitude which is less than or equal to a finite positive value. That is, viii) Summable Signals A sequence is absolutely summable if the sum of the absolute value of all its samples is finite.
A sequence is square summable if the sum of the magnitude squared of all its samples is finite. Classification of Systems ways. Discrete-time systems, like continuous-time systems, can be classified in a variety of i) Linearity A linear system is one which obeys the superposition principle. For a certain system, let the outputs corresponding to inputs x 1 [n] and x 2 [n] are y 1 [n] and y 2 [n] respectively. Now if the input is given by where A and B are arbitrary constants, then the system is linear if its corresponding output is Superposition is a very nice property which makes analysis much simpler. Linearization is a very useful approach to analyzing nonlinear systems. ii) Shift Invariance A shift (or time) invariant system is one that does not change with time. Let a system response to an input x[n] be y[n]. If the input is now shifted by n0 (an integer) samples, then the system is shift invariant if its response to x 1 [n] is We can use the terms linear time-invariant (LTI) and linear shift-invariant interchangeably. iii) Causality The response of a causal system at any time depends only on the input at the current and past instants, not on any future samples. In other words, the output sample y[n o ] for any n o only depends on x[n] for n n0. iv) Stability There are two common criteria for system stability. They are exponential stability and bounded-input bounded-output (BIBO) stability. It requires the response of the system to decay exponentially fast for a finite duration input. The second one merely requires that the output be a bounded sequence if the input is a bounded sequence. Linear Shift-Invariant Systems A discrete-time LTI system, now called as LSI system, like its continuous-time counterpart, is completely characterized by its impulse response. In other words, the impulse response tells us everything we need to know about an LSI system as far as signal processing
is concerned. The impulse response is simply the observed system output when the input is an impulse sequence. The output of the LSI system is using Convolution sum. Condition for Stability Since the impulse response completely characterizes an LSI system, we draw conclusions regarding the stability of a system based on its impulse response. Theorem 1.1. An LSI system is BIBO stable if its impulse response is absolutely summable. Proof: Let the input be bounded, i.e. x[n] < B < 1 for some finite value B. The magnitude of the output is given by So the magnitude of y[n] is bounded if response must be absolutely summable. is finite. In other words, the impulse Condition for Causality Theorem 1.2. An LSI system is causal if and only if its impulse response is a causal sequence. Proof. Consider an LSI system with impulse response h[k]. Two different inputs x1[n] and x2[n] are the same up to a certain point in time, that is x 1 [n] = x 2 [n] for n n0 for some n o. The outputs y 1 [n] and y 2 [n] at n = n o are given by
and Since x 1 [n] = x 2 [n] for n n o, if the system is causal, then the outputs y 1 [n] and y 2 [n] must be the same for n n o. More specifically, y 1 [n o ] = y 2 [n o ]. Now, because for non-negative values of k. Since x 1 [n] may not be equal to x2[n] for n > n o, we must have which means that h[k] = 0 for k < 0. Illustrative Examples: Problem 1: Check whether the system defined by y(n) = x 2 (n) is shift-invariant. Solution: If y(n) = x 2 (n) is the response of the system to x(n), the response of the system to x'(n) = x(n - n o ) is Because y'(n) = y(n - n o ), the system is shift-invariant. Problem 2: Check whether the systems described by the equations i) y(n) = x(n) +x(n - 1 ) ii) y(n) = x(n) + x(n+1) Solution: i) y (n) = x(n) +x(n - 1 ) is causal because the value of the output at any time n = no depends only on the input x(n) at time n o and at time n o - 1. ii) y(n) = x(n) + x(n+1) is noncausal because the output at time n = n o depends on the value of the input at time n o + 1.
Problem 3: An LSI system with unit sample response h(n) = a n u(n) will be stable whenever a < 1. Justify. Solution: We have learnt that the magnitude of y[n] is bounded if impulse response must be absolutely summable. is finite i.e., the Hence, the impulse response is absolutely summable only when a < 1 Problem 4: Prove that the system described by the equation y(n) = n x(n) is not BIBO stable. Solution: Assume that the input signal is a unit step signal x(n) = u(n) which is bounded. u n = 0 n < 0 1 n 0 But the response to a unit step is y(n) = n u(n), which is unbounded as n increases Hence the system is not BIBO stable. Problem 5: Find the response of the system with impulse response h(n) = u(n) for the input x(n)=a n u(n). Solution: With the direct evaluation of the convolution sum we find the response of the system Because u(k) is equal to zero for k < 0 and u(n - k) is equal to zero for k > n, when n < 0, there are no nonzero terms in the sum and y(n) = 0. On the other hand, if n 0, Therefore, Since Summary: Discrete-time signals are essentially a sequence of numbers. Some fundamental sequences, such as the unit impulse, unit step and sinusoidal (both real and complex) sequences are examined because more complex sequences can be expressed as a combination of some or all of these fundamental ones. The most important discrete-time systems are the linear shift invariant (LSI) systems. For these systems, the superposition principle applies which leads to the linear convolution sum. This convolution sum is the way by which we derive the output of the system given an
input. An LSI system is completely characterized by its impulse response. All the properties of such a system is revealed by its impulse response, including causality and BIBO stability. Assignment: Problem 1: Check whether the system described by the equation is shift- varying y(n)=x(n)+x(-n) Problem 2: The input x[n] and output y[n] of a system are linked by the relation y[n] = T{x[n]} = x[- n]. Answer the following questions, giving reasons: a) Is the system linear? (b) Is the system time invariant? c) Is the system causal? Problem 3: Determine which of the following systems are stable: a) y(n) = e x(n) / x(n - 1) b) y(n)=cos(x(n)) Problem 4: A linear shift-invariant system has a unit sample response h(n)=u(-n-1) Find the output if the input is x(n) =3 n u(-n) Problem 5: The input to a linear shift-invariant system is the unit step, x(n) = u(n), and the response is y(n) = δ(n). Find the unit sample response of this system. Simulation: Convolution is a formal mathematical operation, just as multiplication, addition, and integration. Addition takes two numbers and produces a third number, while convolution takes two signals and produces a third signal. Convolution is used in the mathematics of many fields, such as probability and statistics. A convolution of two functions is denoted with the operator *, and is written as Convolution Sum (Discrete functions) Program: clc; clear all; close all; x=input('enter the input'); h=input('enter the impulse response'); %Taking input1 from the user %Taking input2 from the user l=length(x); m=length(h); subplot(3,1,1); stem(0:l-1,x); %Finding lengths %plotting the input sequences
xlabel('n---->'); ylabel('x(n)--->'); title('input sequence'); subplot(3,1,2); stem(0:m-1,h); xlabel('n---->'); ylabel('h(n)--->'); title('impulse response sequence'); %convolution logic y=[]; for n=2:l+m y1=0; for k=1:max(l,m) if l>=m && n-k>0 && n-k<=m y1=y1+ x(k)*h(n-k); elseif m>l && n-k>0 && n-k<=l y1=y1+ h(k)*x(n-k); end end y=[y,y1]; end disp('the convolution result is '); y %Displaying the resultant sequence in command window subplot(3,1,3); %Plotting the resultant sequence stem(0:length(y)-1,y); xlabel('n---->'); ylabel('y(n)=x(n)*h(n)--->'); title('convoluted result'); %Verification of Convolution using Key Function: A= [1 2 3 4] B= [1 1 1 1] C= conv(a, B) % Performing convolution using key function conv Input: Enter the input[1 2 3 4] Enter the impulse response[1 1 1 1] Output: The convolution result is y = 1 3 6 10 9 7 4
References: 1. Digital Signal Processing, Principles, Algorithms and Applications John G Proakis, Dimitris G Manolakis, Pearson Education / PHI, 2007 2. Discrete Time Signal Processing A V Oppenheim and R W Schaffer, PHI, 2009 3. Digital Signal Processing Monson H.Hayes Schaum s Outlines, McGraw-Hill,1999 4. Fundamentals of Digital Signal Processing using Matlab Robert J Schilling, Sandra L Harris, Thomson 2007. 5. Digital Signal processing A Practical Approach, Emmanuel C Ifeachor and Barrie W Jervis, 2 nd Edition, PE 2009 6. Digital Signal Processing A Computer Based Approach, Sanjit K.Mitra, McGraw Hill,2 nd Edition, 2001