LTI DISCRETE-TIME SYSTEM ANALYSIS BY DTFT

9.4 LTI DISCRETE-TIME SYSTEM ANALYSIS BY DTFT

Consider a linear, time-invariant, discrete-time system with the unit impulse response h[n]. We shall find the (zero-state) system response y[n] for the input x[n]. Let

Because

According to Eq. (9.58a) it follows that

This result is similar to that obtained for continuous-time systems. Let us examine the role of H( Ω), the DTFT of the unit impulse response h[n].

Equation (9.63) holds only for BIBO-stable systems and also for marginally stable systems if the input does not contain the system's natural mode(s). In other cases, the response grows with n and is not Fourier transformable. [ † ] Moreover, the input x[n] also has to be DTF transformable. For cases where Eq. (9.63) does not apply, we use the -transform for system analysis.

Equation (9.63) shows that the output signal frequency spectrum is the product of the input signal frequency spectrum and the frequency response of the system. From this equation, we obtain

and

This result shows that the output amplitude spectrum is the product of the input amplitude spectrum and the amplitude response of the system. The output phase spectrum is the sum of the input phase spectrum and the phase response of the system.

We can also interpret Eq. (9.63) in terms of the frequency-domain viewpoint, which sees a system in terms of its frequency response (system response to various exponential or sinusoidal components). The frequency domain views a signal as a sum of various exponential or sinusoidal components. The transmission of a signal through a (linear) system is viewed as transmission of various exponential or sinusoidal components of the input signal through the system. This concept can be understood by displaying the input- output relationships by a directed arrow as follows:

which shows that the system response to e jΩn is H( Ω)e jΩn , and

which shows x[n] as a sum of everlasting exponential components. Invoking the linearity property, we obtain,

which gives y[n] as a sum of responses to all input components and is equivalent to Eq. (9.63) . Thus, X( Ω) is the input spectrum and Y( Ω) is the output spectrum, given by X(Ω)H(Ω).

EXAMPLE 9.13

An LTID system is specified by the equation

Find H( Ω), the frequency response of this system. Determine the (zero-stable) response y[n] if the input x[n] = (0.8) n u[n].

Let x[n] ⇔ X(Ω) and y[n] ⇔ Y(Ω). The DTFT of both sides of Eq. (9.66) yields According to Eq. (9.63)

Also, x[n] = (0.8) n u[n]. Hence,

and

We can express the right-hand side as a sum of two first-order terms (modified partial fraction expansion as discussed in Section B.5-

6 ) as follows: [ † ]

Consequently

According to Eq. (9.34a) , the inverse DTFT of this equation is

This example demonstrates the procedure for using DTFT to determine an LTID system response. It is similar to the Fourier transform method in the analysis of LTIC systems. As in the case of the Fourier transform, this method can be used only if the system is asymptotically or BIBO stable and if the input signal is DTF-transformable. [ † ] We shall not belabor this method further because it is clumsier and more restrictive than the -transform method discussed in Chapter 5 .