4.2. THE DISCRETE-TIME FOURIER TRANSFORM

To see this sampling effect, consider a finite-duration sequence x[n] and construct a periodic signal x ˜[n], for which x[n] is one period   x ˜[n] 0 ≤ n ≤ N − 1 x[n] =  0 otherwise

Discrete-Time Fourier Transform The FT of an discrete-time signal (DTFT) is similarly defined as n=∞ X jω x[n]e−jωn (1) X(e ) =

The periodic signal x ˜[n] has FS coefficients, or the DFT, as 1 X x ˜[n]e−jk(2πn)/N ak = N

n=−∞

then the inverse discrete-time FT is Z 1 X(ejω )ejωn dω x[n] = 2π 2π

(2)

n=

Let ω0 = 2π/N , then these coefficients are scaled samples of the DTFT as

Notes: • The DTFT is periodic with the fundamental period of 2π

ak =

X(ejω ) = X(ej(ω+2π) )

1 X(ejkω0 ) N

• The DTFT is continuous in the frequency domain. This is different from the DFT (Discrete Fourier Transform, or discrete-time Fourier Series), which is discrete in both the time and frequency domains.

The FS representation of x ˜[n] becomes X 1 x ˜[n] = X(ejkω0 )ejkω0 n N k= X 1 X(ejkω0 )ejkω0 n ω0 = 2π

• The DFT is the sampled version of the DTFT in the frequency domain.

As N → ∞, ω0 → 0, and the above sum approaches the integral expression for x[n] in (2).

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k=

1

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Convergence Issues

Examples

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.,

• x[n] = an u[n], |a| < 1 has the DTFT jω

X(e ) = =

∞ X

an u[n]e−jωn

n=−∞ ∞ X

(ae−jω )n =

n=0

∞ X

1 . 1 − ae−jω

|x[n]| < ∞

n=−∞

or

 1, |n| ≤ N 1 • The rectangular pulse s[n] = 0, |n| > N1

∞ X

|x[n]|2 < ∞

n=−∞

has the DTFT

jω

X(e ) =

N1 X

e−jωn = . . .

n=−N1

sin(ω(N1 + 12 )) = . sin(ω/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 2π

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• The periodic impulse train of period N, P∞ x[n] = k=−∞ δ[n − kN ] has Fourier coefficients ak = N1 for all k (check!) and so has the DTFT,

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] = e

jω0 n

jω

⇔ X(e ) =

∞ X

∞ X

¶ 2πk 2πak δ ω − X(e ) = N k=−∞ ¶ µ ∞ 2πk 2π X δ ω− = N N jω

2πδ(w − w0 − 2πl)

l=−∞

If we consider a periodic sequence x[n] with period N and Fourier series representation X ak ej(2π/N )n x[n] =

µ

k=−∞

k=

then its DTFT is given by, jω

X(e ) =

∞ X

k=−∞

2πak δ(ω −

2πk ). N

Examples: • The periodic signal x[n] = cos(ω0 n) = 12 ejω0 n + 12 e−jω0 n , with ω0 = 2π 5 has the DTFT, for −π ≤ ω < π, ¶ µ ¶ µ 2π 2π + πδ ω + , X(ejω ) = πδ ω − 5 5 which repeats periodically with period 2π. ES156 – Harvard SEAS

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This is the duality between the discrete time Fourier transform and the continuous time Fourier series. Notice the similarities: Z 1 x[n] = X(ejω )ejωn dω 2π 2π ∞ X jω x[n]e−jωn X(e ) =

Properties of the DTFT The DTFT has properties analogous to the continuous-time FT. A few notable differences are as follows. ¡ ¢ ¡ ¢ • Periodicity: X ejω = X ej(ω+2π) • Differencing in time: F

n=−∞

¡

x[n] − x[n − 1] ←→ 1 − e • Accumulation: n X

x[k]

¢ −jω

¡

X e

compared to

¢ jω

x(t) =

←→

k=−∞

+πX(ej0 )

∞ X

1 ak = T

1 |x[n]| = 2π n=−∞

Z

T

2π

• Linearity:

¯ ¯ ¯X(ejω )¯2 dω

ax1 [n] + bx2 [n] ↔ aX1 (ejω ) + bX2 (ejω )

Note that all the energy is contained in one period of the DTFT. ¡ ¢ • Duality: Since X ejω is periodic, it has FS representation with coefficients given by x[−n]. ES156 – Harvard SEAS

x(t)e−jkω0 t dt

Other properties that are similar to the CTFT:

• Parseval’s relation: 2

Z

This property can be used to ease the computation of certain Fourier series coefficients.

δ(ω − 2πk)

k=−∞

∞ X

ak ejkω0 t

k=−∞

¡ jω ¢ 1 X e 1 − e−jω

F

∞ X

• Time and Frequency shifting: x[n − n0 ] ↔ e−jωn0 X(ejω ) ejω0 n x[n] ↔ X(ej(ω−ω0 ) ) 7

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1 ↔ Y (e ) = 2π

• Conjugation and Conjugate Symmetry:

jω

x∗ [n] ↔ X ∗ (e−jω ) jω

Ev{x[n]} ↔

X1 (ejθ )X2 (ej(ω−θ) dθ 2π

Example: System input x[n] = (α)n u[n] and impulse response h[n] = (β)n u[n] lead to output y[n] = x[n] ∗ h[n], then by taking the DTFTs, we see that

jω

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

Z

Re{X(ejω )

Od{x[n]} ↔ jIm{X(ejω )

Y (ejω ) = X(ejω )H(ejω ) 1 1 · −jω 1 − βe 1 − αe−jω α/(α − β) −β/(α − β) = + 1 − αejω 1 − βejω α β (α)n u[n] − (β)n u[n] ⇒ y[n] = α−β α−β for α 6= β

• Time reversal: x[−n] ↔ X(e−jω ).

=

8