Lecture 9 ELE 301: Signals and Systems

Prof. Paul Cuff

Princeton University

Fall 2011-12

Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 1 / 16 Discrete-time Fourier Transform

Represent a Discrete-time signal using δ functions

Properties of the Discrete-time Fourier Transform

I Periodicity I Time Scaling Property I Multiplication Property

Periodic Discrete Duality

DFT

Constant-Coefficient Difference Equations

Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 2 / 16 Fourier Transform for Discrete-time Signals

Z x[n] = X (f )ei2πfndf , 1 ∞ X X (f ) = x[n]e−i2πfn. n=−∞

X (f ) is always periodic with period 1.

Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 3 / 16 Continuous-representation of a discrete-time signal

∞ X x(t) , x[k]δ(t − k). k=−∞

Notice that

∞ ! Z ∞ X F[x(t)] = x[k]δ(t − k) e−i2πft dt −∞ k=−∞ ∞ X Z ∞ = x[k]δ(t − k)e−i2πft dt k=−∞ −∞ ∞ X = x[k]e−i2πfk dt. k=−∞

Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 4 / 16 Properties of the Discrete-time Fourier Transform

Inherits properties from continuous-time. Easy Properties:

Linearity Conjugation Convolution = Multiplication in frequency domain Parseval’s Theorem (integrate over one period) Time shift

Properties that require care:

Time-scaling Multiplication (circular convolution in frequency)

Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 5 / 16 Time-scaling

In continuous time we can scale by an arbitrary real number. In discrete-time we scale only by integers. For an integer k, define

 x[n/k] if n is a multiple of k, x [n] = k 0 if n is not a multiple of k.

xk [n] ⇔ X (kf ).

Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 6 / 16 Compressing in time

Compressing in time requires decimation.

Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 7 / 16 Discrete Difference

What is the Fourier transform of y[n] = x[n] − x[n − 1]?

Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 8 / 16 Dual Derivative Formula

The dual to the continuous-time differentiation formula still holds.

i nx[n] ⇔ X 0(f ). 2π

Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 9 / 16 Accumulation

Pn What is the Fourier transform of y[n] = m=−∞ x[m]? (Hint: Inverse of discrete difference)

Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 10 / 16 Parseval’s Theorem

Theorem: ∞ Z 1/2 X 2 2 Ex = |x[n]| = |X (f )| df . n=−∞ −1/2

Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 11 / 16 Multiplication Property

Multiplication in time equates to circular convolution in frequency.

Notice that multiplying δ functions is not well defined.

This is dual to what we saw in the Fourier series.

Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 12 / 16 Discrete Periodic Duality

periodic signal = discrete transform (though not integer f ) discrete signal = periodic transform

Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 13 / 16 Discrete Fourier Transform Notice that a discrete and periodic signal will have a discrete and periodic transform. This is convenient for numerical computation (computers and digital systems). The DFT is (almost) equivalent to the discrete-time Fourier series of the periodic extension. For period N, let  x[0]   x[1]  x =    .   .  x[N − 1]

Then   a0  a1  DFT [x] =    .   .  aN−1

whereCuffak (Lectureare the 9) Fourier seriesELE coefficients.301: Signals and Systems Fall 2011-12 14 / 16 The DFT Matrix

Since the Fourier transform is linear, the DFT can be encompassed in a matrix.

DFT [x] = Fx.

Matlab uses the fast-Fourier-transform algorithm to compute the DFT (using the fft command).

Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 15 / 16 Constant-Coefficient Difference Equations

n M X X ak y[n − k] = bk x[n − k]. k=0 k=0

Find the Fourier Transform of the impulse response (the transfer function of the system, H(f )) in the frequency domain.

Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 16 / 16