Signals and Systems - Electrical Engineering

Problems 507

8.18. FIR filter—MATLAB
An FIR filter has a nonrecursive input–output relation

y[n]=

∑^5

k= 0

kx[n−k]

(a) Find and plot using MATLAB the impulse responseh[n]of this filter.

(b) Is this a causal and stable filter? Explain.

(c) Find and plot the unit-step responses[n]for this filter.

(d) If the inputx[n]for this filter is bounded, i.e.,|x[n]|< 3 , what would be a minimum boundMfor the

output (i.e.,|y[n]|≤M)?

(e) Use the MATLAB functionfilterto compute the impulse responseh[n]and the unit-step responses[n]

for the given filter and plot them.

8.19. LTI and convolution sum—MATLAB
The impulse response of a discrete-time system ish[n]=(−0.5)nu[n].
(a) If the input of the system isx[n]=δ[n]+δ[n−1]+δ[n−2], use the linearity and time invariance of
the system to find the corresponding outputy[n].
(b) Find the convolution sum corresponding to the above input, and show that your solution coincides
with the outputy[n]obtained above.
(c) Use the MATLAB functionconvto find the outputy[n]due to the given inputx[n]. Plotx[n],h[n], and
y[n]using MATLAB.

8.20. Steady state of IIR systems—MATLAB
Suppose an IIR system is represented by a difference equation

y[n]=ay[n−1]+x[n]

wherex[n]is the input andy[n]is the output.

(a) If the inputx[n]=u[n]and it is known that the steady-state response isy[n]= 2 , what would beafor

that to be possible (in steady statex[n]= 1 andy[n]=y[n−1]= 2 sincen→∞).

(b) Writing the system input asx[n]=u[n]=δ[n]+δ[n−1]+δ[n−2]+··· then according to the

linearity and time invariance, the output should be

y[n]=h[n]+h[n−1]+h[n−2]+···

Use the value forafound above, that the initial condition is zero (i.e.,y[−1]= 0 ) and that the input is

x[n]=u[n], to find the values of the impulse responseh[n]forn≥ 0 using the above equation. The

system is causal.

(c) Use the MATLAB functionfilterto compute the impulse responseh[n]and compare it with the one

obtained above.

8.21. Causal systems and real-time processing
Systems that operate under real-time conditions need to be causal—that is, they can only process present
and past inputs. When no real-time processing is needed the system can be noncausal.