8.2 Discrete-Time Signals 465
Solution
The even component ofx[n] is given by
xe[n]=0.5
(
x[n]+x[−n]
)
Whenn=0 thenxe[0]=0.5× 2 x[0]=4, whenn>0 thenxe[n]=0.5x[n], and whenn<0 then
xe[n]=0.5x[−n], giving
xe[n]=
2 +0.5n − 4 ≤n≤− 1
4 n= 0
2 −0.5n 1 ≤n≤ 4
0 otherwise
The odd component
xo[n]=0.5
(
x[n]−x[−n]
)
gives 0 whenn=0, 0.5x[n] forn>0, and−0.5x[−n] whenn<0, or
xo[n]=
− 2 −0.5n − 4 ≤n≤− 1
0 n= 0
2 −0.5n 1 ≤n≤ 4
0 otherwise
The sum of these two components givesx[n]. n
RemarksExpansion and compression of discrete-time signals is more complicated than in the continuous-
time signals. In the discrete domain, expansion and compression can be related to the change of the sampling
period in the sampling. Thus, if a continuous-time signal x(t)is sampled using a sampling period Ts, by chang-
ing the sampling period to MTsfor an integer M> 1 , we obtain fewer samples, and by changing the sampling
period to Ts/L for an integer L> 1 , we increase the number of samples. For the corresponding discrete-time
signal x[n], increasing the sampling period would give x[Mn], which is called thedownsamplingof x[n]by
M. Unfortunately, because the argument of discrete-time signals must be integers, it is not clear what x[n/L]
is unless the values for n are multiples of L (i.e., n=±0,±L,± 2 L,...) without a clear definition when n
takes other values. This leads to the definition of theupsampledsignal
xu[n]=
{
x[n/L] n=±0,±L,± 2 L,...
0 otherwise
(8.15)
To replace the zero entries with the values obtained by decreasing the sampling period we need to low-pass filter
the upsampled signal. MATLAB provides the functionsdecimateandinterpto implement the downsampling
and upsampling without losing information due to possible frequency aliasing. In Chapter 10, we will continue
the discussion of these operations including their frequency characterization.
8.2.4 Basic Discrete-Time Signals
The representation of discrete-time signals via basic signals is simpler than in the continuous-time
domain. This is due to the lack of ambiguity in the definition of the impulse and the unit-step
discrete-time signals. The definitions of impulses and unit-step signals in the continuous-time
domain are more abstract.
