616 C H A P T E R 10: Fourier Analysis of Discrete-Time Signals and Systems
10.4.2 DFT of Aperiodic Discrete-Time Signals
We obtain the DFT of an aperiodic signaly[n] by sampling its DTFT,Y(ejω), in frequency. Suppose
we choose{ωk= 2 πk/L,k=0,...,L− 1 }as the sampling frequencies, where an appropriate value
for the integerL>0 needs to be determined. Analogous to the sampling-in-time we did before,
sampling-in-frequency generates a periodic signal in time:
y ̃[n]=
∑∞
r=−∞
y[n+rL] (10.47)
Now, ify[n] is of finite lengthN, then whenL≥Nthe periodic expansion ̃y[n] clearly displays a first
period equal to the given signaly[n] (with some zeros attached at the end whenL>N). On the other
hand, if the lengthL<Nthe first period ofy ̃[n] does not coincide withy[n] because of superposition
of shifted versions of it (this corresponds totime aliasing, the dual of frequency aliasing, which occurs
in time sampling).
Assumingy[n] is of finite lengthNand thatL≥N, as the dual of sampling in time we then have that
̃y[n]=
∑∞
r=−∞
y[n+rL]⇔ Y[k]=Y(ej^2 πk/L)=
N∑− 1
n= 0
y[n]e−j^2 πnk/L k=0,...,L− 1 (10.48)
The equation on the right is the DFT ofy[n]. The inverse DFT is the Fourier series representation of
̃y[n] (normalized with respect toL) or its first period
y[n]=
1
L
L∑− 1
k= 0
Y[k]ej^2 πnk/L 0 ≤n≤L− 1 (10.49)
whereY[k]=Y(ej^2 πk/L).
Thus, instead of the frequency aliasing that sampling-in-time causes, we have time-aliasing whenever
the lengthNofy[n] is greater than the chosenLin the sampling-in-frequency. In practice, the gen-
eration of the periodic extensiony ̃[n] is not needed—we just need to generate a period that either
coincides withy[n] whenL=N, or whenL>Nthat coincides withy[n] with a sequence ofL−N
zeros attached to it (i.e.,y[n] ispadded with zeros). To avoid time aliasing we do not consider choosing
L<N.
If the signaly[n] is a very long signal, in particular ifN→∞, it does not make sense to compute
its DFT, even if we could. Such a DFT would give the frequency content of the whole signal and
since an infinite-length signal could have all types of frequencies its DFT would just give no valuable
information. A possible approach to obtain, over time, the frequency content of a signal with a large
time support is to window it and compute the DFT of each of these segments. Thus, wheny[n] is of
infinite length, or its length is much larger than the desired or feasible lengthL, we use a window
WL[n] of lengthL, and representy[n] as the superposition
y[n]=
∑
m
ym[n] where ym[n]=y[n]WL[n−mL] (10.50)