1

Discrete Time Fourier Transform

Maryam Mahsal Khan (Lecturer)»B.Sc (CSE) – UET Peshawar»M.Sc(EE) – UTP Malaysia

Representation of Sequences by Fourier Transforms

Many sequences can be represented by a Fourier integral of the form as

Frequency response of a LTI system is simply the Fourier transform of the impulse response.

nj

n

j enxeX ωω −∞

−∞=∑= ][)(.)(

21][ ωπ

ωπ

π

ω deeXnx njj∫−=

nj

n

j enheH ωω −∞

−∞=∑= ][)(.)(

21][ ωπ

ωπ

π

ω deeHnh njj∫−=

The frequency response of discrete-time LTI system is always a periodic function with period 2π.

integer. an for ),()( generally, More )2( reHeH jrj ωπω =+

)(][][)( )2()2( ωωπωπω j

n

nj

n

njj eHenhenheH === ∑∑∞

−∞=

−∞

−∞=

+−+

Representation of Sequences by Fourier Transforms Fourier Transform (Convergence)

Determining the class of signals that can be represented Fourier transform is equivalent to considering the convergence of the infinite sum of the Fourier transform.A sufficient condition for convergence can be found as

Thus, if a sequence is absolutely summable, then its Fourier transform exists. The series can be shown to converge uniformly to a continuous function of ω.Since a stable sequence is, by definition, absolutely summable, all stable sequences have Fourier transforms.

∞<≤≤= ∑∑∑∞

−∞=

−∞

−∞=

−∞

−∞= n

nj

n

nj

n

j nxenxenxeX ][][][)( ωωω

2

Fourier Transform (Interpretation)

Signals: The Fourier Transform of a signal x[n] describes the frequency content of the signal.

At each frequency , the magnitude spectrum describes the amount of that frequency contained in the signal.At each frequency , the phase spectrum describes the location (relative shift) of that frequency component of the signal.

Systems: The frequency response of a linear system describes how frequencies input to the system are modified:

An input frequency component is amplified or attenuated by a factorAn input frequency component is shifted by an amount

)( ωjeX

0ω )( 0ωjeX

0ω )( 0ωjeX∠

)( ωjeH

0ω.)( 0ωjeH

0ω).( 0ωjeH∠

Example 2.19 Ideal Frequency-Selective Filters: Ideal Lowpass filter

DTFT of Ideal Low-Pass Filter Contd..

Convergence of the Fourier TransformThe oscillatory behaviour – Gibbs Phenomena

( ) ωω

πω j

M

Mn

cjM e

nneH −

−=∑=

sin

3

Frequency-Domain Representation of Discrete-Time Systems

Eigenfunctions for LTI systemConsider an input sequence

The corresponding output of a LTI discrete-time system with impulse response is

If we define

Then we have

We define as an eigenfunction of the system, and the associated eigenvalue is .

,for ][ ∞<<∞−= nenx njω

][nh

.][][][*][][ )( ⎟⎠

⎞⎜⎝

⎛=== ∑∑

∞

−∞=

−∞

−∞=

−

k

kjnj

k

knj ekheekhnhnxny ωωω

,][)( ∑∞

−∞=

−=k

kjj ekheH ωω

.)(][ njj eeHny ωω=

nje ω

)( ωjeH

Frequency-Domain Representation of Discrete-Time Systems (Cont’d)

Frequency response of the system is defined as

Real and imaginary representation

Magnitude and phase polar representation

,][)( ∑∞

−∞=

−=k

kjj ekheH ωω

)()()( ωωω jI

jR

j ejHeHeH +=

.)()( )( ωωω jeHjjj eeHeH ∠=

Example 2.17 Frequency Response of the Ideal Delay

Consider the ideal delay system defined by

If we consider as input to this system.Then we have the output

Therefore, the frequency response of the ideal delay is

Real and imaginary representation

Magnitude and phase representation

integer. fixeda is where],[][ dd nnnxny −=njenx ω=][

njnjnnj eeeny dd ωωω −− == )(][

.)( dnjj eeH ωω −=

).sin()cos()( ddj njneH ωωω +=

.)( and 1)( djj neHeH ωωω −=∠=

Example 2.20 Frequency Response of the Moving-Average System

The impulse response of a moving-average system is

The frequency response is

⎪⎩

⎪⎨⎧ ≤≤−

++=otherwise.0

,1

1][ 21

21

MnMMMnh

∑−=

−

++=

2

11

1)(21

M

Mn

njj eMM

eH ωω

ω

ωω

j

MjMj

eee

MM −

+−

−−

++=

111 )1(

21

21

.)2/sin(

)2/)1(sin(1

1 2/)(21

21

12 MMjeMMMM

−−++++

= ω

ωω

4

Example 2.20 Frequency Response of the Moving-Average System (Cont’d)

?)()( 00 * ωω jj eHeH =−

Application Example - Signal Smoothing

A common DSP application is the removal of noise from a signal corrupted by additive noise. A simple 3-point moving average algorithm is given by:

])1[][]1[(31][ +++−= nxnxnxny

0 5 10 15 20 25 30 35 40 45 50-2

0

2

4

6

8

Time index n

Ampl

itude

d[n]s[n]x[n]

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

Time index n

Ampl

itude

y[n]s[n]

% Signal Smoothing by Averagingclf;% Generate random noise, d[n]R = 51; d = 0.8*(rand(R,1) - 0.5); % Generate uncorrupted signal, s[n]m = 0:R-1; s = 2*m.*(0.9.^m); % Generate noise corrupted signal, x[n]x = s + d'; subplot(2,1,1);plot(m,d','r-',m,s,'g--',m,x,'b-.');xlabel('Time index n');ylabel('Amplitude');legend('d[n] ','s[n] ','x[n] ');% do smoothingx1 = [0 0 x];x2 = [0 x 0];x3 = [x 0 0];y = (x1 + x2 + x3)/3;subplot(2,1,2);plot(m,y(2:R+1),'r-',m,s,'g--');legend( 'y[n] ','s[n] ');xlabel('Time index n');ylabel('Amplitude');

Symmetry Properties of the Fourier Transform

Symmetry properties of the Fourier transform are often very useful for simplifying the solution of problems.Some basic definitions

Conjugate-symmetric sequenceConjugate-antisymmetric sequenceAny sequence can be expressed as a sum of a conjugate-symmetric and conjugate-antisymmetric sequence

Even sequence (real): Odd sequence (real):

][][ * nxnx ee −=

][][][ 0 nxnxnx e +=

][][ * nxnx oo −−=

[ ][ ] ][][][

21][

][][][21][

**

**

nxnxnxnx

nxnxnxnx

oo

ee

−−=−−=

−=−+=

][][ nxnx ee −=

][][ nxnx oo −−=

Symmetry Properties of the Fourier Transform (Cont’d)

Similarly, a Fourier transform can be decomposed into a sum of conjugate-symmetric and conjugate-antisymmetric functions.

For real sequences, the real part of the Fourier transform is an even function, and the imaginary part is an odd function.

)( ωjeX

)()()( ωωω jo

je

j eXeXeX +=

[ ] )()()(21)( ** ωωωω j

ejjj

e eXeXeXeX −− =+=

])[*][(21]}[Re{ nxnxnx += ])[*][(

21]}[Im{ nxnxnxj −=

[ ] )()()(21)( ** ωωωω j

ojjj

o eXeXeXeX −− −=−=

5

Fourier Transform Theorems

Frequency Response of LCCDE Example: Determining an h[n] from Difference Equation

[ ] [ ] [ ] [ ]1411

21

−−=−− nxnxnyny

[ ] [ ]121

41

21][

211

41

211

1)(

211411

)(

411)(

21)(

1

−⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

−−

−=

−

−=

−=−

−

−

−

−

−

−

−−

nununh

TablesTransformFrom

e

e

eeH

e

eeH

eeHeeH

nn

j

j

j

j

j

j

j

jjjj

ω

ω

ω

ω

ω

ω

ω

ωωωω

6

Example 2.22 Determining the Impulse Response from the Frequency Response

The frequency response of a high-pass filter with delay

⎩⎨⎧

<<<

=−

.0

)(c

cnj

jde

eHωω

πωωωω

),())(1()( ωωωωωω jlp

njnjjlp

njj eHeeeHeeH ddd −−− −=−=

⎩⎨⎧

≤<<

=.,0

,1)(

πωωωωω

c

cjlp eH

nnnh c

lp πωsin][ =

)()(sin][][

d

dcd nn

nnnnnh−−

−−=πωδ