Signals and Systems - Electrical Engineering

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

Downsampling and Upsampling

Although the expanding and contracting of discrete-time signals is not as obvious as in the

continuous time, the dual effects of contracting and expanding in time and frequency also occur

in the discrete case. Contracting and expanding of discrete-time signals relates to downsampling and

upsampling.

Downsamplinga signalx[n] means getting rid of samples (i.e., contracting the signal). The signal

downsampled by an integer factorM>1 is given by

xd[n]=x[Mn] (10.14)

If x[n] has a DTFT X(ejω),−π/M≤ω≤π/M, and zero otherwise in [−π,π)(analogous to

bandlimited signals in continuous time), by replacingnbyMnin the inverse DTFT,x[n] gives

x[Mn]=

1

2 π

π/∫M

−π/M

X(ejω)ejMnωdω=

1

2 π

∫π

−π

1

M

X(ejρ/M)ejnρdρ

where we letρ=Mω. Thus, the DTFT ofxd[n] isM^1 X(ejω/M)—that is, an expansion by a factor ofM

of the DTFT ofx[n].

Upsamplinga signalx[n], on the other hand, consists in addingL−1 zeros for some integerL>1 in

between its samples—that is, the upsampled signal is

xu[n]=

{

x[n/L] n=0,±L,± 2 L,...

0 otherwise

(10.15)

thus expanding the original signal. The DTFT of the upsampled signal,xu[n], is found to be

Xu(ejω)=X(ejLω) −π≤ω < π

Indeed, the DTFT ofxu[n] is

Xu(ejω)=

∑

n=0,±L,...

x[n/L]e−jωn=

∑∞

m=−∞

x[m]e−jωLm=X(ejLω) (10.16)

indicating that it is a contraction of the DTFT ofx[n].

n A signalx[n], bandlimited toπ/Min[−π,π)or|X(ejω)|= 0 ,|ω|> π/Mfor an integerM> 1 , can be

downsampled to generate a discrete-time signal

xd[n]=x[Mn] with Xd(ejω)=

1

M

X(ejω/M) (10.17)

which is an expanded version ofX(ejω).

n A signalx[n]is upsampled to generate a signalxu[n]=x[n/L]forn=±kL,k=0, 1, 2,...,and zero

otherwise. The DTFT ofxu[n]isX(ejLω), or a compressed version ofX(ejω).