Lab 2: Linear Time-Invariant Systems

In this experiment, you will study the output response of linear time-invariant (LTI) systems using MATLAB, and learn how to use MATLAB to implement the convolution sum. You will also investigate the properties of LTI systems. The objective of this experiment is: (1) to study how to compute the output of LTI systems, and (2) to study the properties of discrete-time LTI systems.

1. Introduction to Linear Time-Invariant (LTI) Systems In discrete time, linearity provides the ability to completely characterize a system in terms of its response [pic] to a signal of the form [pic] for all [pic]. If a linear system is also time-invariant, then the responses [pic] will become [pic]. The combination of linearity and time-invariance therefore allows a system to be completely described by its impulse response [pic]. The output of the system [pic] is related to the input [pic] through the convolution sum as follows: [pic] Similarly, the output [pic] of a continuous-time LTI system is related to the input [pic] and the impulse response [pic] through the following convolution integral: [pic] The convolution of discrete-time sequences [pic] and [pic] represented mathematically by the expression given in [pic] can be viewed pictorially as the operation of flipping the time axis of the sequence [pic] and shifting it by [pic] samples, then multiplying [pic] by[pic] and summing the resulting product sequence over [pic]. This picture arises from the properties of linearity and time-invariance for discrete-time systems. The signal [pic] can be constructed from a linear superposition of delayed and scaled impulses, i.e. [pic] [pic] Since an LTI system can be represented by its response to a single impulse, the output of an LTI system corresponds to the superposition [pic]. Mathematically, this results in the convolution sum. It can also prove that (1) is equivalent to the following equation: [pic] [pic]

2. Computing the Output of an LTI System Using MATLAB In this section, you will learn how to use the MATLAB function conv to compute the output of LTI systems. The MATLAB function conv computes the convolution sum. Assuming that [pic] and [pic] are finite-length sequences. If [pic] is nonzero on the interval[pic]and [pic] is nonzero only on the interval [pic], then y[n] can be nonzero only on the interval [pic] [pic] This means that conv need only compute y[n] for the [pic] samples on this interval. If x is an [pic]-dimensional vector containing x[n] on the interval [pic] and h is an [pic]-dimensional vector containing h[n] on the interval [pic], then y = conv(h, x) returns in y the [pic] samples of y[n] on the interval in (5). However, conv does not return the indices of samples of y[n] stored in y, which makes sense because the intervals of x and h are not input to conv. a) Consider the finite-length signal, [pic] Analytically determine [pic], where [pic] denotes the convolution operation. b) Compute the nonzero samples of [pic] using conv. Plot your results using stem. The following code may help you: nx = 0:5; %interval of [pic] x = [1 1 1 1 1 1]; % [pic] figure, stem(nx, x) %plot [pic] y = conv(x, x); % [pic] ny = 0 : ? ; % interval of [pic], you should set a value for “?” figure, stem(ny, y) % plot [pic] What is the length of y[n]? Does the plot agree with the signal determined in (a)? c) Consider the finite-length signal, [pic] Analytically determine [pic]. Next, using conv to compute [pic]. Plot your results using stem. Is the system causal? d) Now consider the signal [pic], where[pic]. Compute [pic] using conv. Plot your result using stem. How does [pic] compare to the signal [pic] derived in (c)? Is this system causal? e) Since the MATLAB function conv does not keep track of the time indices of the sequences that are convolved, some extra book keeping is required to determine the proper indices for the result of the conv function. For the sequences:
[pic]
construct vectors h and x with proper indices. Define [pic] and compute y = conv(h,x). Determine the proper time indexing for y and plot the result. The following code may help you. x = [0, 1, 0, 1, 0]; % [pic] nx = –1 : 3; % interval of [pic] h = [2, 0, –2]; % [pic] nh = 0 : 2; -1:1 % interval of [pic] y = conv(x, h); % convolution ny = ?: ?; % interval of y, you must assign values to the “?”s figure, stem(ny, y) % plot [pic] f) Verify your result in (e) analytically. g) Consider the following signal and impulse response, [pic] If you wish to compute the convolved output using y = conv(h,x), you must deal appropriately with the infinite length of both h[n] and x[n]. Store in the vector x the value of x[n] for[pic]and store in the vector h the values of h[n] for [pic]. Now calculate y = conv(h,x) using MATLAB. Since you have truncated both [pic] and [pic], argue that only a portion of the output of y will be valid. Specify which values in the output are valid and which are not.

3. Properties of Discrete-Time LTI Systems In this section, you will verify the commutative, associative and distributive properties of convolution for a specific set of signals. In addition, you will examine the implications of these properties for series and parallel connections of LTI systems. a) Consider the following three signals: [pic] [pic] [pic] Define the MATLAB vector x to represent [pic] on the interval [pic], and the vectors h1 and h2 to represent [pic] and [pic] for [pic]. Also define nx to be an appropriate index vector for x[n] and nh for h1[n] and [pic]. Make appropriately labeled plots of all the signals using stem. b) The commutative property states that the result of a convolution is the same regardless of the order of the operands, i.e. [pic] . (6) This implies that, if [pic]is the input and [pic] is the impulse response, the output is equivalent to that of an LTI system with impulse response [pic] and input [pic]. Use conv with h1 and x to verify this property. Is the output of conv the same regardless of the order of the input arguments? c) Convolution is also distributive. This means that [pic]. (7) This implies that the output of two LTI systems connected in parallel is the same as one system whose impulse response is the sum of the impulse responses of the parallel systems. Figure 1 illustrates this property.

[pic] Figure 1 Distributive property of convolution Verify the distributive property using x, h1 and h2. Compute the sum of the outputs of the two LTI systems with impulse responses [pic] and [pic] when [pic] is the input. Compare this with the output of the LTI system whose impulse response is [pic] when the input is [pic]. Do these two methods of computing the output give the same results? d) Convolution also possesses the associative property, i.e., [pic] (8) This property implies that the result of processing a signal with a series of LTI systems is equivalent to processing the signal with a single LTI system whose impulse response is the convolution of all the individual impulse responses of the connected systems. Figure 2 illustrates this property for the case of two LTI systems connected in series.

[pic]

Figure 2 Associative property of convolution

Use the following steps to verify the associative property using x, h1 and h2: • Let w[n] be the output of the LTI system with impulse response [pic], as shown in Fig. 2. Compute w[n] by convolving [pic] and [pic]. • Compute the output [pic] of the whole system by convolving [pic] with [pic]. • Find the impulse response [pic]. • Convolve [pic] with[pic]to get the output [pic]. Compare [pic] and [pic]. Are they the same?

4. Linearity and Time-Invariance In this section, you will become more familiar with the system properties of linearity and time-invariance. An important property of discrete-time LTI systems is that if [pic] is the response to the unit impulse [pic], then the response of the system [pic] to any [pic] is determined by the convolution sum. An analogous property holds for continuous-time LTI systems. Consider the systems System 1: [pic] System 2: [pic] a) Consider these three input signals: [pic] [pic] [pic] For System 1, store in w1, w2, and w3 the responses to these three inputs. Use stem to plot the four functions represented by w1, w2, w3 and w1+2*w2. The following code may help you. x1 = [1 0 0 0 0 0]; % define [pic] x2 = [0 1 0 0 0 0]; % define [pic] x3 = [1 2 0 0 0 0]; % define [pic] h = ?; % impulse response of [pic] w1 = conv(h, x1); % response of [pic] w2 = conv(h, x2); % response of [pic] w3 = conv(h, x3); % response of [pic] nw = 0 : ?; % interval of [pic] figure, stem(nw, w1) % plot [pic] figure, stem(nw, w2) % plot[pic] figure, stem(nw, w3) % plot [pic] figure, stem(nw, w1+2*w2) % plot [pic] b) Is System 1 linear? Is the system time-invariant? c) Repeat (a) for System 2. Is System 2 linear? Is it time-invariant? The following code may help you. ny = 0 : 5; % interval of [pic] y1 = cos(x1); % response of [pic] y2 = cos(x2); % response of [pic] y3 = cos(x3); % response of [pic]