**Department of Communication & Electronics Engineering** Faculty of Engineering Philadelphia University Signal Analysis & ProcessingDiscrete -Time LTI SystemsLecture (26) Lecturer: Ibrahim Abu-Isbeih

**h[n]** d[n] LTI System Figure 3.20 3.6 Discrete-Time LTI Systems: In this section, we develop the fundamental input/output relationship for discrete-time, linear, time-invariant (LTI) systems. It is useful to characterize an LTI system in terms of its impulse response. The impulse response of an LTI system is defined as the output of the LTI system due to a unit impulse signal input applied at time t=0 as illustrated in Figure 3.20. i.e., if x[n]=d [n] then y[n]=h[n]. Signal Analysis & Processing

**y[n]** x[n] LTI System Figure 3.21 The impulse response of the LTI system shown in Figure 3.20 is denoted by h[n], whereh[n] is the response tod[n]. Consider a discrete-time LTI system with input x[n] shown in Figure 3.21, if the impulse response of the LTI system is h[n] then the output of this LTI system is given by: Signal Analysis & Processing

**Hence, the impulse response of the LTI system h[n] is the** only function needed to characterize the system completely with respect to its input and output. The output of an LTI system with impulse response h[n] is then given by the convolution sum: Utilizing the commutivity property of convolution, we can also write the above equation as: Signal Analysis & Processing

**The procedure of evaluating the convolution sum (the LTI** system output) was discussed in session two. For example, the output of a discrete-time LTI system with impulse response h[n]=u[n] to the input x[n]=3u[n-2] is given by: Signal Analysis & Processing

**x[n]** h[n] 3 2 1 1 1 n n -1 3 0 2 1 -1 3 0 2 1 Example 3.6.1:Find the output of a discrete-time LTI system with impulse response h[n]=u[n]-u[n-2] to the input Solution:The output of the system is given by: To find the above convolution sum: - Graph x[n] and h[n]: * Signal Analysis & Processing

**x[k]** h[k] 3 2 1 1 1 k k -1 3 0 2 1 -1 3 0 2 1 • Replace n with kin x[n] and h[n] • Graph h[n-k] h[n-k] 1 1 1 k n-1 n n-2 1 0 Signal Analysis & Processing

**x[k]** 3 2 k -1 3 0 2 1 According to the convolution sumconvolution can be divided into the following cases: • 1. For n< 0: The functions x[k] and h[n-k] • do not overlap, then y[n]=0 h[n-k] 1 1 1 k n-1 n n-2 1 0 Signal Analysis & Processing

**x[k]** 3 2 k -1 3 0 2 1 • 2. For n =0: h[n-k] 1 1 1 k -2 1 -1 0 Signal Analysis & Processing

**x[k]** 3 2 k -1 3 0 2 1 • 3. For n =1: h[n-k] 1 1 1 k -1 1 0 Signal Analysis & Processing

**x[k]** 3 2 k -1 3 0 2 1 • 4. For n =2: h[n-k] 1 1 1 k 0 2 1 Signal Analysis & Processing

**x[k]** 3 2 k -1 3 0 2 1 • 5. For n =3: h[n-k] 1 1 1 k 0 1 2 3 Signal Analysis & Processing

**x[k]** 3 2 k -1 3 0 2 1 • 6. For n ≥ 4: h[n-k] 1 1 1 k 0 1 n-1 n n-2 2 Signal Analysis & Processing

**Then the output of the system is :** y[n] 5 5 3 2 n -1 4 0 1 3 2 Signal Analysis & Processing

**s[n]** u[n] LTI System Figure 3.22 Step-Response: In this lecture we introduce the step-responseof discrete-time LTI systems. The step-response of an LTI system with impulse response h[n] is defined as the output of the LTI system due to the unit step signal input as illustrated in Figure 3.22. i.e., if x[n]=u[n] then y[n]=s[n]=u[n]*h[n]. Signal Analysis & Processing

**h[n]** 5 5 5 5 ……… 2 2 2 2 n 0 1 2 4 -1 3 5 6 Example 3.6.2: Find the step-response of an LTI system with impulse response h[n]=2u[n]+3u[n-4]. Signal Analysis & Processing

**Solution:** To find the step-response of the given system, let x[n]=u[n] then the step-response is: Signal Analysis & Processing

**h[n]** 3 1 n 4 -2 -1 3 0 2 1 -1 Example 3.6.3:Find the step-response of an LTI system with impulse response Signal Analysis & Processing

**Solution:The step-response of the system is:** wherex[n]=u[n]. Using the convolution properties we have: Signal Analysis & Processing