Discrete-Time Systems in the Time-Domain
Background A discrete-time system processes an input signal in the time-domain to generate an output signal with more desirable properties by applying an algorithm composed of simple operations on the input signal and delayed versions. Systems here refer to the processes between the input and output signals. Objectives 1. To illustrate the simulation of some simple discrete-time systems on the computer using SCILAB. 2. To investigate the time-domain properties of SCILAB. Theories T2.1 For a linear discrete-time system, if
y1 n and y2 n are the responses to the input
sequences x1 n and x2 n , respectively, then for an input x n x1 n x2 n , the response is given by y n y1 n y2 n . The superposition property of the response y n must hold for any arbitrary constants α and β and for all possible inputs x1 n and x2 n . Otherwise, if y n does not hold for at least one set of nonzero values of α and β, or one set of nonzero input sequences x1 n and x2 n , then the system is said to be non-linear. T2.2 For a time-invariant discrete-time system, if y1 n is the response to an input x1 n , then the
response to an input x n x1 n n0 is simply y n y1 n n0 . Where n0 is any positive or negative
integer. The above relation between the input and output must hold for any arbitrary input sequence and its corresponding output. If it does not hold for at least one input sequence and its corresponding output sequence, the system is said to be time-varying. T2.3 A linear time-invariant (LTI) discrete-time system satisfies both the linearity and the timeinvariance properties. T2.4 If
inputs u1 n and u2 n , respectively, then u1 n u2 n for n N .
y1 n and y2 n are
for n N also implies that y1 n y2 n
T2.5 A discrete-time system is said to be bounded-input, bounded-output (BIBO) stable if, for any bounded input sequence x n , the corresponding output y n is also a bounded sequence, that is, if x n Bx for all values of n , then the corresponding output y n is also bounded, that is, y n B y for all values of n , where Bx and By are finite constants.
Digital Signal Processing Laboratory Using SCILAB/SCICOS (RU Espina ©2009)
The response of a digital filter to a unit sample sequence n is called the unit sample response
or, simply, the impulse response, and denoted as h n . Correspondingly, the response of a discrete-time system to a unit step sequence n , denoted as s n , is its unit step response or, simply the step response. T2.7 The response y n of a linear, time-invariant discrete-time system characterized by an impulse k
response h n to an input signal x n is given by y n written as y n
h k x n k ,
which can be alternately
convolution sum of the sequences x n and h n , and is represented compactly as: y n h n x n , where the notation denotes the convolution sum. T2.8 The overall impulse response h n of the LTI discrete-time system obtained by a cascade
h n k x k
by a simple change of variables. The sum just shown is called the
connection of two LTI discrete-time systems with the impulse responses h1 n and h2 n , respectively, and of Figure 2.1 are such that h1 n h2 n n , then the LTI system h2 n is said to be the inverse of the LTI system h1 n and vice versa. as shown in Figure 2.1, is given by h n h1 n h2 n . If the two LTI systems in the cascade connection
h1 n h2 n
Figure 2.1 Discrete-Time System An...
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