Signals and Systems - Electrical Engineering

10.2 Discrete-Time Fourier Transform 575

According to the formula for the DTFT atω=0, we have that

X(ej^0 )=

∑∞

n=−∞

x[n]ej^0 n=

∑∞

n=−∞

α|n|=

2

1 −α

− 1 =

1 +α

1 −α

and according to Equation (10.8), equivalently we have

X(ej^0 )=

1 −α^2

1 − 2 α+α^2

=

1 −α^2

( 1 −α)^2

=

1 +α

1 −α n

10.2.2 Duality in Time and Frequency

In practice, there are many signals of interest that do not satisfy the absolute summability condition,

and so we cannot find their DTFTs with the definition given in the previous section. Duality in the

time and frequency representation of signals permits us to obtain the DTFT of those signals.

Consider the DTFT of the signalδ[n−k] for some integerk. SinceZ[δ[n−k]]=z−kwith ROC the

wholez-plane except for the origin, the DTFT ofδ[n−k] ise−jωk. By duality, as in the continuous-

time domain, we would expect that the signale−jω^0 n,−π≤ω 0 < π, would have 2πδ(ω+ω 0 )(where

δ(ω)is the analog delta function) as its DTFT. Indeed, the inverse DTFT of 2πδ(ω+ω 0 )gives

1

2 π

∫π

−π

2 πδ(ω+ω 0 )ejωndω=e−jω^0 n

∫π

−π

δ(ω+ω 0 )dω=e−jω^0 n

Using these results, we have the following dual pairs:

∑∞

k=−∞

x[k]δ[n−k]⇔

∑∞

k=−∞

x[k]e−jωk

∑∞

k=−∞

X[k]e−jωkn⇔

∑∞

k=−∞

2 πX[k]δ(ω+ωk) (10.9)

The top left equation is the generic representation of a discrete-time signalx[n] and the corresponding

term on the right is its DTFTX(ejω), one more verification of Equation (10.1). The bottom pair is a

dual of the above.^1 Using Equation (10.9), we then have the following dual pairs as special cases:

x[n]=Aδ[n]⇔X(ejω)=A

y[n]=A,−∞<n<∞⇔Y(ejω)= 2 πAδ(ω) −π≤ω < π

The signaly[n] is not absolutely summable, and as a constant it does not change from−∞to∞,

so that its frequency isω=0, thus its DTFTY(ejω)is concentrated in that frequency. Consider then

(^1) Calling this a “dual” is not completely correct given that theωkare discrete values of frequency instead of continuous as expressed by
ω, and that the delta functions are not the same in the continuous and the discrete domains, but a duality of some sort exists in these
two pairs, which we would like to take advantage of.