ECE 8443 – Pattern RecognitionEE 3512 – Signals: Continuous and Discrete

• Objectives:Derivation of the DTFTTransforms of Common SignalsProperties of the DTFTLowpass and Highpass Filters

• Resources:Wiki: Discete-Time Fourier TransformJOS: DTFT DerivationMIT 6.003: Lecture 9CNX: DTFT PropertiesSKM: DTFT Properties

LECTURE 16: DISCRETE-TIME FOURIER TRANSFORM

Audio:URL:

EE 3512: Lecture 16, Slide 2

Discrete-Time Fourier Series• Assume x[n] is a discrete-time periodic signal. We want to represent it as a

weighted-sum of complex exponentials:

Note that the notation <N> refers to performing the summation over an N samples which constitute exactly one period.

• We can derive an expression for the coefficients by using a property of orthogonal functions (which applies to the complex exponential above):

• Multiplying both sides by and summing over N terms:

• Interchanging the order of summation on the right side:

• We can show that the second sum on the right equals N if k = r and 0 if k r:

otherwise,0

...,2,,0,)/2( NNkNe

Nn

nNjk

Nn Nk

nNrkjk

Nn Nk

nNjrnNjkk

Nn

nNjr eceecenx )/2)(()/2()/2()/2(

TececnxNk

njkk

Nk

nTjkk /2, 0

)/2( 0

nNjre )/2(

Nn

nNrkj

Nkk

Nn

nNjr ecenx )/2)(()/2(

Nn

nNjkk enx

Nc )/2(1

EE 3512: Lecture 16, Slide 3

Discrete-Time Fourier Series (Cont.)• This provides a closed-form expression for the Discrete-Time Fourier Series:

• Note that the DT Fourier Series coefficients can be interpreted as in the CT case – they can be plotted as a function of frequency (k0) and demonstrate the a periodic signal has a line spectrum.

• As expected, this DT Fourier Series (DTFS) obeys the same properties we have seen for the CTFS and CTFT.

• Next, we will apply the DTFS to nonperiodic signals to produce the Discrete-Time Fourier Transform (DTFT).

(analysis)11

)(synthesis/2,

0

0

)/2(

0)/2(

Nn

njk

Nn

nNjkk

Nk

njkk

Nk

nTjkk

enxN

enxN

c

Tececnx

EE 3512: Lecture 16, Slide 4

Periodic Extension• Assume x[n] is an aperiodic,

finite duration signal.

• Define a periodic extension of x[n]:

• Note that:

• We can apply the complex Fourier series:

• Define: . Note this is periodic in with period 2.

• This implies: . Note that these are evenly spaced samples of

our definition for .

22

],[~ Nn

Nnxnx

][~~ nxNnx

Nasnxnx ][~

njk

n

njkN

Nn

njkN

Nnn

njkN

Nkk

enxN

enxN

enxN

c

Tecnx

00

2

1

0

0

][1

][~1

][~1

/2,~

2/

2/

0

2/

2/

nj

n

j enxN

eX

][

1

01 jk

n eXN

c

jeX

EE 3512: Lecture 16, Slide 5

• Derive an inverse transform:

• As

• This results in our Discrete-Time Fourier Transform:

Notes:

• The DTFT and inverse DTFT are not symmetric. One is integration over a finite interval (2π), and the other is summation over infinite terms.

• The signal, x[n] is aperiodic, and hence, the transform is a continuous function of frequency. (Recall, periodic signals have a line spectrum.)

• The DTFT is periodic with period 2. Later we will exploit this property to develop a faster way to compute this transform.

The Discrete-Time Fourier Transform

0000000

2

1)

2(

2

1)

1(~

njk

N

Nk

jknjkN

Nk

jknjkN

Nk

jk eeXN

eeXeeXN

nx

dN

nxnxN 00 ,02

],[][~,

equation)(analysis][

equation)(synthesis2

1][

2

n

njj

njj

enxeX

deeXnx

EE 3512: Lecture 16, Slide 6

Example: Unit Pulse and Unit Step

• Unit Pulse:

The spectrum is a constant (and periodic over the range [-,].

• Shifted Unit Pulse:

Time delay produces a phase shift linearly proportional to . Note that these functions are also periodic over the range [-,].

• Unit Step:

Since this is not a time-limited function, it hasno DTFT in the ordinary sense. However, it can be shown that the inverse of this function isa unit step:

1][][

][][

n

nj

n

njj enenxeX

nnx

0

0

0

00

0

and1

][][

][][

neeXeeX

eennenxeX

nnnx

njjnjj

nj

n

nj

n

njj

][][ nunx

)(1

1

j

j

eeX

EE 3512: Lecture 16, Slide 7

Example: Exponential Decay

• Consider an exponentially decaying signal:

1][][ anuanx n

j

n

nj

n

njn

n

njj

ae

aeeaenxeX

1

1

)(][00

2

2222222

)cos(21

1

Hence,

)cos(21)(sin)(cos)cos(21))sin(())cos(1(

:Note

aaeX

aaaaaaa

j

22 ))sin(())cos(1(

1

)sin())cos(1(

1

1

1

aa

jaaaeeX

jj

EE 3512: Lecture 16, Slide 8

The Spectrum of an Exponentially Decaying Signal

a

a

aa

aaeX

a

a

aa

aaeX

j

j

1

1

)1(

1

)21(

1

)cos(21

11

1

)1(

1

)21(

1

)0cos(21

1

2

2

2

2

2

20

Lowpass Filter:

Highpass Filter:

EE 3512: Lecture 16, Slide 9

Finite Impulse Response Lowpass Filter

1

11

1

,0

,1

,0

][

Nn

NnN

Nn

nh

)2/sin(

))2/1(sin()( 1

1

1

1

1

NeeeX

N

Nn

njN

Nn

njj

• The frequency response ofa time-limited pulse is alowpass filter.

• We refer to this type offilter as a finite impulse response (FIR) filter.

• In the CT case, we obtaineda sinc function (sin(x)/x) for the frequency response. This is close to a sinc function, and is periodic with period 2.

EE 3512: Lecture 16, Slide 10

Example: Ideal Lowpass Filter (Inverse)

c

cn

ndenh cnj

sin)1(

2

1][

The Ideal Lowpass Filter Is Noncausal!

EE 3512: Lecture 16, Slide 11

Properties of the DTFT

• Periodicity:

• Linearity:

• Time Shifting:

• Frequency Shifting:

Example:

CTFT)thefrom(different)2( jj eXeX

jj ebYeaXnbynax ][][

jnj eXennx 00

)( 00 jnj eXnxe

][)1(][ nxnxeny nnj

Note the roleperiodicityplays in theresult.

EE 3512: Lecture 16, Slide 12

Properties of the DTFT (Cont.)

• Time Reversal:

• Conjugate Symmetry:

Also:

• Differentiation inFrequency:

• Parseval’s Relation:

• Convolution:

][][

jeX

d

djnnx

jj eXeXnx *real][

functionsoddareImand

functionsevenareReand

jj

jj

eXeX

eXeX

oddandimaginarypurelyisodd,andrealis][

evenandrealiseven,andrealis][

j

j

eXnx

eXnx

jeXnx ][

deXnx j

n

2

2

2

2

1][

)()()(][*][][ jjj eXeHeYnhnxny

EE 3512: Lecture 16, Slide 13

Properties of the DTFT (Cont.)

• Time-Scaling:)(

otherwise0

kofmultipleaisnif]/[][ jk

k eXknx

nx

EE 3512: Lecture 16, Slide 14

Example: Convolution For A Sinewave

EE 3512: Lecture 16, Slide 15

Summary• Introduced the Discrete-Time Fourier Transform (DTFT) that is the analog of

the Continuous-Time Fourier Transform.

• Worked several examples.

• Discussed properties: