discrete-time signals & systems -...

Discrete-time Signals & SystemsFrequency Analysis of Discrete-Time Signals

S Wongsa

11

S Wongsa

Dept. of Control Systems and Instrumentation Engineering,

KMUTT

Overview

Discrete-time processing of continuous-time signals

Time & Frequency Domain

Sampling Theorem

Aliasing

Frequency Analysis of Discrete-Time Signals

22

DTFT

DFT


Signals & Systems

A signal is a varying phenomenon that can be measured.

A system responses to particular signals by producing other signals.

33Source: 6.003 Signals & Systems, MIT, Fall 2009.

Signals & Systems

An image is also a signal!

44Source: Yao Wang, Introduction, Review of Signals & Systems, Image Quality Metrics, Polytechnic University, Brooklyn, NY


55

Slowly time-varying signals tend to have low frequency content, while signals

With abrupt change in their amplitudes have high frequency content.

Demo: fft.exe

Signals in the Time and Frequency Domain

The sequence “I owe you” (IOU)

66Source: U. Karrenberg, An interactive multimedia Introduction to Signal Processing, 2nd Edition, Springer.

The word “history”


77


88


Sampling Reconstruction

e.g. DSP,

Controller etc.

99

• Most of the signals in the physical world are CT signals, e.g. voltage &

current, pressure, temperature, velocity, etc.

• But digital computations are done in discrete time.


1010Source: Prof. Mark Fowler, EECE 301 Signals & Systems, Binghamton University.

Sampling

Sampling is the process of getting a discrete signal from a continuous one.

It enables the processing of signal by digital computer.

T

)(tx )(txs

1111

• Discrete-time / sampled signal

K,2,1,0 ],[)()( ±±=== nnxnTxtxs

T

where T is a sampling time.

Sampling

)(tx

)(txs

)(tTδX

1212

)()()( ttxtx Ts δ=where

∑∞

−∞=

−=n

T nTtt )()( δδ

T

Sampling in frequency domain

The Fourier transform of : )(txs

∑∞

−∞=

−=k

ss kXT

X )(1

)( ωωω

where

Ts

πω

2= is the sampling frequency in rad/sec

If x(t) has bandwidth B and if B2>ω

)(ωX is the Fourier transform of x(t)

1313

If x(t) has bandwidth B and if Bs 2>ω

Sampling theorem

A bandlimited signal with bandwidth B can be reconstructed completely and

exactly from its samples as long as they are taken at rate .Bs 2>ω

• is called the Nyquist sampling frequency / Nyquist rate.Bs 2=ω

1414

Aliasing

Q : What if the signal is not bandlimited nor is the sampling frequency greater

than the Nyquist sampling frequency?

A : The high frequency components of x(t) will be transposed to low-frequency

components, leading to a phenomenon called aliasing.

1515

Aliasing

• What are the consequences of aliasing?

- a distorted version of the original signal x(t).

Example: SSUM – samplexpo & aliasexpo

1

2

3

Amplitude

1616

See also Demonstration 2: Sampling - http://www.youtube.com/watch?v=OQNR099y8mM

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-3

-2

-1

0

1

Time (sec)

Amplitude

Aliasing: a 52 Hz sinusoid sampled at 50 Hz.

Anti-Aliasing Filter

To avoid aliasing, in practice we use a CT lowpass filter before the ADC to restrict the

bandwidth of a signal to approximately satisfy the sampling theorem.

Fs = 44.1 kHz

1717Source: Prof. Mark Fowler, EECE 301 Signals & Systems, Binghamton University.

Overview



Sampling Theorem

Aliasing


1818

DTFT

DFT


Recall : Sampling Analysis

)(ωX

1919

Alternative Way to Compute

0)()(tj

eXttxωω −→−

2020

0)()( 0

tjeXttx

ωω −→−

1)( →tδ&

CTFT vs. DTFT

2121

DTFT – Discrete-Time Fourier Transform

Frequency spectra of DT signals

• DTFT

∑∞

−∞=

Ω−=Ωn

njenxX ][)(

• Inverse DTFT

ΩΩ= Ω∫ deXnx nj

ππ2

)(2

1][

2222

- DTFT of a discrete-time signal is a function of a continuum of frequencies.

- DTFT is complex valued.

)()()( Ω∠Ω=Ω XjeXX

DTFT – Discrete-Time Fourier Transform

Periodicity: DTFT is a periodic function of Ωwith period of 2π.

Symmetry: By replacing Ω by -Ω we can see that

)(][)( Ω==Ω− ∑∞

Ω XenxX nj

2323

)(][)( Ω==Ω− ∑−∞=

Ω XenxXn

nj

Even symmetry of |X(Ω)|

Odd symmetry of ∠X(Ω)

It is sufficient to consider the DTFT over only Ω=[0 π].

DTFT

EXAMPLE: DTFT of an exponential function

][5.0][ nunx n= n

n

j

n

njn eeX )5.0(5.0)(00

∑∑∞

=

Ω−∞

=

Ω− ==Ω

r

rrr

qqq

qn

n

−−

=+

=∑

1

1212

1

+Ω−−=Ω

qjeX

)5.0(1lim)(

12

2424

Ω−∞→ −−

=Ωjq

e

eX

5.01

)5.0(1lim)(

2

Ω−−=Ω

jeX

5.01

1)(

π/ΩNB: Normalized frequency = radians/unit time

DTFT

EXAMPLE: DTFT of a rectangular pulse

2525N = 2q + 1 = no. of non-zero data

Common DTFT Pairs

2626Source: E.W. Kamen & B.S. Heck, Fundamentals of Signals & Systems using Web and MATLAB, 2007.

Properties of DTFT

2727

NB: Convolution sum is defined as

][*][][][][][][*][][ nxnhkhknxknhkxnhnxnykk

=−=−== ∑∑∞

−∞=

∞

−∞=

Can DTFT easily be implemented?

• DTFT is not suitable for implementing on a digital computer because

- Any signal can be measured only on a finite number of points.

- We are able to compute the spectrum only at specific discrete values of Ω.

∑∞

−∞=

Ω−=Ωn

njenxX ][)(

2828

- We are able to compute the spectrum only at specific discrete values of Ω.

• We can approximate DTFT of a finite signals at N points by sampling

the spectrum in frequency Discrete-Fourier Transform (DFT))(ΩX ][kX

DFT – Discrete Fourier Transform

NkXkX /2|)(][ π=ΩΩ=

16-point DFT

Frequency spacing

2929

Hz rad/sec; ;rad/sample 2

N

F

NN

ssωπFrequency spacing =


The DFT pair:

• DFT

∑−

=

−=

1

0

2

][][N

n

knNj

enxkX

π

• Inverse DFT

1,...,1,0 −= Nk

3030

∑−

=

=1

0

2

][1

][N

k

knNj

ekXN

nx

π

1,...,1,0 −= Nn


16-point DFT

3131

• Periodicity: The DFT spectrum is a periodic function of k with period N.


EXAMPLE: Computation of DFT

Suppose that and for all other integers n.

Find the DFT of this sequence.

,1]3[,2]2[,2]1[,1]0[ ==== xxxx 0][ =nx

34

22

4

2

4

2

3

0

2

]3[]2[]1[]0[

][][

kjkjkj

n

knNj

exexexx

enxkX

πππ

π

−−−

=

−

+++=

=∑

3232

34

22

4

2

4

2

34

244

221

]3[]2[]1[]0[

kjkjkj

kjkjkj

eee

exexexx

πππ−−−

−−−

+++=

+++=

For k = 0:

61221 ]0[ =+++=X



34

22

4

2

4

2

221 ][kjkjkj

eeekXπππ

−−−+++=

For k = 1:

jj

eeeXjjj

+−+−+=

+++=−−−

)()1(2)(21

221]1[3

4

22

4

2

4

2 πππ

3333

j

jj

−−=

+−+−+=

1

)()1(2)(21

For k = 2:

0

)1()1(2)1(21

221]2[6

4

24

4

22

4

2

=

−++−+=

+++=−−−

πππjjj

eeeX



34

22

4

2

4

2

221 ][kjkjkj

eeekXπππ

−−−+++=

For k = 3:

eeeXjjj

+++=−−−

221]3[9

4

26

4

23

4

2 πππ

3434

j

jj

eeeX

+−=

−+−++=

+++=

1

)()1(2)(21

221]3[

=+−

=

=−−

=

=

3,1

2,0

1,1

0,6

][

kj

k

kj

k

kX


EXAMPLE: Computation of Inverse DFT

=+−

=

=−−

=

=

3,1

2,0

1,1

0,6

][

kj

k

kj

k

kX

Consider the signal in the previous Example with DFT given by

3535

=+− 3,1 kj

Evaluate the inverse DFT

+−+−−+=

+++=

= ∑=

34

2

4

2

34

22

4

2

4

2

3

0

2

)1()1(6 4

1

]3[]2[]1[]0[4

1

][4

1][

njnj

njnjnj

k

knNj

ejej

eXeXeXX

ekXnx

ππ

πππ

π



+−+−−+=

34

2

4

2

)1()1(64

1 ][

njnj

ejejnxππ

For n = 0:

[ ] 4/4)1()1(6 4

1]0[

=

=+−+−−+= jjx

3636

1 =

For n = 1:

[ ][ ]2

4/84/)1()1(6

4/))(1())(1(6

4/)1()1(6 ]1[3

4

2

4

2

=

=++−+=

−+−+−−+=

+−+−−+=

jj

jjjj

ejejxjj

ππ



+−+−−+=

34

2

4

2

)1()1(64

1 ][

njnj

ejejnx

ππ

For n = 2:

4/)1()1(6 ]2[6

4

22

4

2

+−+−−+= ejejx

jjππ

3737

For n = 3:

[ ][ ]1

4/44/)1()1(6

4/))(1())(1(6

4/)1()1(6 ]3[9

4

23

4

2

=

=−−++−+=

+−+−−−+=

+−+−−+=

jj

jjjj

ejejxjj

ππ

[ ][ ]2

4/84/)1()1(6

4/)1)(1()1)(1(6

=

=−+++=

−+−+−−−+=

jj

jj

Some Properties of DFT

3838

NB: If x[n] and y[n] are two N-point periodic sequences, the circular convolution is defined as,

∑−

=

−=⊗1

0

][][][][N

m

mnymxnynx

( )

−=⊗ ∑

−

=

1

0

][][1

][][1 N

l

lkYlXN

kYkXN

⊗

⊗

Discrete Fourier Transform

=

=n

qnnx

other all ,0

2,....,2,1,0 ,1][

x[n]

EXAMPLE : DTFT and DFT of a pulse

3939

...

0 1 2 3 2q-1 2q n

1



4040



N=22Ω−

Ω

Ω+=Ω jqe

q

X)2/sin(

)2

1(sin

)(

q=5

4141

N=88

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