9-dtft.pdf

7/29/2019 9-dtft.pdf

1/6

4.2. THE DISCRETE-TIME FOURIER

TRANSFORM

Discrete-Time Fourier Transform

The FT of an discrete-time signal (DTFT) is similarly

defined as

X(ej

) =

n=n=

x[n]ejn

(1)

then the inverse discrete-time FT is

x[n] =1

2

2

X(ej)ejnd (2)

Notes:

The DTFT is periodic with the fundamental

period of 2

X(ej) = X(ej(+2))

The DTFT is continuous in the frequency

domain. This is different from the DFT (DiscreteFourier Transform, or discrete-time Fourier

Series), which is discrete in both the time and

frequency domains.

The DFT is the sampled version of the DTFT in

the frequency domain.

ES156 Harvard SEAS 1

To see this sampling effect, consider a finite-duration

sequence x[n] and construct a periodic signal x[n], forwhich x[n] is one period

x[n] =

x[n] 0 n N 1

0 otherwise

The periodic signal x[n] has FS coefficients, or theDFT, as

ak =1

N

n=

x[n]ejk(2n)/N

Let 0 = 2/N, then these coefficients are scaled

samples of the DTFT as

ak =1

NX(ejk0)

The FS representation of x[n] becomes

x[n] =

k=

1

NX(e

jk0

)e

jk0n

=

k=

1

2X(ejk0)ejk0n0

As N , 0 0, and the above sum approaches

the integral expression for x[n] in (2).


7/29/2019 9-dtft.pdf

2/6

Examples

x[n] = anu[n], |a| < 1 has the DTFT

X(ej) =

n=

anu[n]ejn

=

n=0

(aej

)n

=

1

1 aej .

The rectangular pulse s[n] =

1, |n| N1

0, |n| > N1

has the DTFT

X(ej) =N1

n=N1

ejn = . . .

=sin((N1 +

12

))

sin(/2).

This is the discrete-time counterpart to the sinc

function, which appears as the Fourier Transform

of the rectangular pulse. Unlike the

continuous-time counterpart, the discrete-time

sinc function is periodic with period 2


Convergence Issues

Convergence of the infinite summation in the analysis

equation is guaranteed if either x[n] is absolutely

summable, or if the sequence has finite energy, i.e.,

n=

|x[n]| <

or

n=

|x[n]|2 <


7/29/2019 9-dtft.pdf

3/6

Fourier Transform for Periodic Signals

The DTFT of a periodic signal is a periodic impulse

train in the frequency domain. First, by substitution

into the synthesis equation, we see that

x[n] = ej0n X(ej) =

l=

2(w w0 2l)

If we consider a periodic sequence x[n] with period N

and Fourier series representation

x[n] =

k=

akej(2/N)n

then its DTFT is given by,

X(ej) =

k=

2ak( 2k

N).

Examples:

The periodic signal

x[n] = cos(0n) = 12ej0n + 1

2ej0n, with

0 =25

has the DTFT, for < ,

X(ej) =

2

5

+

+

2

5

,

which repeats periodically with period 2.


The periodic impulse train of period N,

x[n] =

k= [n kN] has Fourier coefficientsak =

1N for all k (check!) and so has the DTFT,

X(ej) =

k=

2ak

2k

N

=2

N

k=

2k

N


7/29/2019 9-dtft.pdf

4/6

Properties of the DTFT

The DTFT has properties analogous to the

continuous-time FT. A few notable differences are as

follows.

Periodicity: X

ej

= X

ej(+2)

Differencing in time:

x[n] x[n 1]F

1 ej

X

ej

Accumulation:n

k=

x[k]F

1

1 ejX

ej

+X(ej0)

k=

( 2k)

Parsevals relation:

n=

|x[n]|2 =1

22X(ej)2 d

Note that all the energy is contained in one

period of the DTFT.

Duality: Since X

ej

is periodic, it has FS

representation with coefficients given by x[n].


This is the duality between the discrete time

Fourier transform and the continuous timeFourier series. Notice the similarities:

x[n] =1

2

2

X(ej)ejnd

X(ej) =

n=

x[n]ejn

compared to

x(t) =

k=

akejk0t

ak =1

TT

x(t)ejk0tdt

This property can be used to ease the

computation of certain Fourier series coefficients.

Other properties that are similar to the CTFT:

Linearity:

ax1[n] + bx2[n] aX1(ej) + bX2(ej)

Time and Frequency shifting:

x[n n0] ejn0X(ej)

ej0nx[n] X(ej(0))


7/29/2019 9-dtft.pdf

5/6

Conjugation and Conjugate Symmetry:

x[n] X(ej)

And if x[n] is real, this means X(ej) = X(ej).

Furthermore,

Ev{x[n]} Re{X(ej)

Od{x[n]} jI m{X(ej)

Time reversal: x[n] X(ej).

Time expansion: x(k)[n] =

8

9-dtft.pdf

Documents