Signals and Systems - Electrical Engineering

Problems 503

or equivalently that

( 1 −α)

∑∞

n= 0

αn= 1

provided|α|< 1.

(c) Verify your results by using symbolic MATLAB to find an expression for the above sum.

8.3. Periodicity of sampled signals—MATLAB
Consider an analog periodic sinusoidx(t)=cos( 3 πt+π/ 4 )being sampled using a sampling periodTsto
obtain the discrete-time signalx[n]=x(t)|t=nTs=cos( 3 πTsn+π/ 4 ).
(a) Determine the discrete frequency ofx[n].
(b) Choose a value ofTsfor which the discrete-time signalx[n]is periodic. Use MATLAB to plot a few
periods ofx[n], and verify its periodicity.
(c) Choose a value ofTsfor which the discrete-time signalx[n]is not periodic. Use MATLAB to plotx[n]
and choose an appropriate length to show the signal is not periodic.
(d) Determine under what condition the value ofTsmakesx[n]periodic.

8.4. Even and odd decomposition and energy—MATLAB
Suppose you sample the analog signal

x(t)=

{

1 −t 0 ≤t≤ 1

0 otherwise

with a sampling periodTs=0.25to generatex[n]=x(t)|t=nTs.

(a) Use MATLAB to plotx[−n]for an appropriate interval.

(b) Findxe[n]=0.5[x[n]+x[−n]]and plot it carefully using MATLAB.

(c) Findxo[n]=0.5[x[n]−x[−n]]and plot it carefully using MATLAB.

(d) Verify thatxe[n]+xo[n]=x[n]graphically.

(e) Compute the energy ofx[n]and compare it to the sum of the energies ofxe[n]andxo[n].

8.5. Signal representation in terms ofu[n]—MATLAB
We have shown how any discrete-time signal can be represented as a sum of weighted and shifted versions
ofδ[n]. Given that

δ[n]=u[n]−u[n−1]

it should be possible to represent any signal as a combination of unit-step functions. Consider a discrete-

time rampr[n]=nu[n], which in terms ofδ[n]is written as

r[n]=

∑∞

k=−∞

r[k]δ[n−k]

Replacer[k]=ku[k]and useδ[n]=u[n]−u[n−1]to show thatr[n]can be expressed in terms ofu[n]as

r[n]=

∑∞

k= 1

u[n−k]

Does this equation make sense? Use MATLAB to plot the obtainedr[n]to help you answer this.