Signals and Systems - Electrical Engineering

618 C H A P T E R 10: Fourier Analysis of Discrete-Time Signals and Systems

above sequence and obtainLvalues corresponding to the DFT ofx[n] of lengthL(why this could

be seen as a better version of the DFT ofx[n] is discussed below in frequency resolution).

n Noncausal aperiodic signals:When the given signalx[n] is noncausal of lengthN—that is, the

samples

{x[n],n=−n 0 ,..., 0, 1,...,N−n 0 − 1 }

are given—we need to recall that a periodic extension ofx[n] orx ̃[n] was used to obtain its DFT.

This means that we need to create a sequence ofNvalues corresponding to the first period of

̃x[n]—that is,

x[0]x[1]···x[N−n 0 −1]

︸ ︷︷ ︸

causal samples

x[−n 0 ]x[−n 0 +1]···x[−1]

︸ ︷︷ ︸

noncausal samples

where as indicated the samplesx[−n 0 ]x[−n 0 +1]···x[−1] are the values that makex[n] non-

causal. If we wish to consider zeros afterx[N−n 0 −1] to be part of the signal, so as to obtain a

better DFT transform as we discuss later in frequency resolution, we simply attach zeros between

the causal and noncausal components—that is,

x[0]x[1]···x[N−n 0 −1]

︸ ︷︷ ︸

causal samples

0 0···0 0x[−n 0 ]x[−n 0 +1]···x[−1]

︸ ︷︷ ︸

noncausal samples

to compute anL>NDFT of the noncausal signal. The periodic extensionx ̃[n] represented

circularly instead of linearly would clearly show the above sequence.

n Periodic signals:If the signalx[n] is periodic of periodNwe will then chooseL=N(or a multiple

ofN) and calculate the DFTX[k] by means of the FFT algorithm. If we use a multiple of the period

(e.g.,L=MNfor some integerM>0), we need to divide the obtained DFT by the valueM. For

periodic signals we cannot chooseLto be anything but a multiple ofNas we are really computing

the Fourier series of the signal. Likewise, no zeros can be attached to a period (or periods when

M>1) to improve the frequency resolution of its DFT—by attaching zeros to a period we distort

the signal.

n Frequency resolution:When the signalx[n] is periodic of periodN, the DFT values are normalized

Fourier series coefficients ofx[n] that only exist for the harmonic frequencies{ 2 πk/N}, as no fre-

quency components exist for any other frequencies. On the other hand, whenx[n] is aperiodic,

the number of possible frequencies depend on the lengthLchosen to compute its DFT. In either

case, the frequencies at which we compute the DFT can be seen as frequencies around the unit cir-

cle in thez-plane. In both cases one would like to have a significant number of frequencies in the

unit circle so as to visualize the frequency content of the signal well. The number of frequencies

considered is related to thefrequency resolutionof the DFT of the signal.

n If the signal is aperiodic we can improve the frequency resolution of its DFT by increasing the

number of samples in the signal without distorting the signal. This can be done bypadding

the signal with zeros(i.e., attaching zeros to the end of the signal). These zeros do not change

the frequency content of the signal (they can be considered part of the aperiodic signal) but

permit us to increase the available frequency components of the signal.