lecture 2 signals & systems review marc moonen & toon van waterschoot dept. e.e./esat,...

Lecture 2

Signals & Systems Review

Marc Moonen & Toon van Waterschoot

Dept. E.E./ESAT, K.U.Leuven

[email protected]

www.esat.kuleuven.be/scd/

DSP-IIVersion 2009-2010 Lecture-2: Signals & Systems Review p. 2

Signals & Systems Review

• Discrete-Time/Digital Signals sampling, quantization, reconstruction

• Discrete-Time SystemsLTI, impulse response, convolution, z-transform, frequencyresponse, frequency spectrum, IIR/FIR

• Discrete/Fast Fourier Transform(I)DFT, FFT


Introduction: Digital Signal Processing?

• Signal = a physical quantity that varies as a function of some independent variable(s), e.g. time, position, frequency, …– 1-dimensional: speech signal, audio signal,

electromagnetic/radio signal, …– 2-dimensional: image, …– N-dimensional

• Processing = `filtering’ (mostly) = noise reduction, equalization, signal separation, …

• Digital = …in `digital domain’

her

e…



Analog Signal

Processing Circuit

Analog Domain (Continuous-Time Domain)

Analog signal processing

Analog IN Analog OUT

)(tu

dtetufU tfj ..2).()(

(=Spectrum/Fourier Transform)

)(ty

dtetyfY tfj ..2).()(

Jose

ph

Fo

uri

er (

1768

-183

0)



Analog-to-

Digital

Conversion

DSP

Digital-to-

Analog

ConversionAnalog IN Analog OUTDigital IN

Digital OUT

0110100101

1001100010

Digital signal processing in an analog world

Analog domain

Digital

domainAnalog domain


Discrete-Time/Digital Signals 1/10

Analog-to-

Digital

Conversion

DSP

Digital-to-

Analog

ConversionAnalog IN Analog OUTDigital IN

Digital OUT

0110100101

1001100010

sampling& quantization

reconstruction


k

sD Tkttxtx ).().()(

Discrete-Time/Digital Signal 2/10 : Sampling

• Time-domain samplingamplitude amplitude

discrete-time [k]continuous-time (t)

impulse train

).(][ sTkxkx )(tx

sT

It will turn out (page 27) that a spectrum can be computed from x[k], which (remarkably) will be equal to the spectrum (Fourier transform) of the (continuous-time) sequence of impulses =

discrete-time

signal

continuous-time

signal

0 1 2 3 4


Discrete-Time/Digital Signals 3/10 : Sampling

• Spectrum replication– time domain:

– frequency domain:

magnitude

frequency (f)

magnitude

frequency (f)

k

sD Tkttxtx ).().()(

k ss

D T

kfX

TfX )(.

1)(

)( fX )( fX D



• Sampling theorem– analog signal spectrum X(f) has a bandwidth of fmax Hz

– spectrum replicas are separated by fs =1/Ts Hz

– no spectral overlap if and only if

magnitude

frequency



• Sampling theorem– terminology:

• sampling frequency/rate fs

• Nyquist frequency fs/2

• sampling interval/period Ts

– e.g. CD audio: fs = 44,1 kHz

• Anti-aliasing prefilters– if then frequencies above the Nyquist

frequency are ‘folded’ into lower frequencies (=aliasing)– to avoid aliasing, sampling is usually preceded by a low-

pass (=anti-aliasing) filtering

Har

ry N

yqui

st (

7 fe

brua

ri 18

89 –

4 a

pril

1976

)


Discrete-Time/Digital Signals 6/10 : Quantization

• B-bit quantization

amplitude

discrete time [k]

0Q

2Q3Q

-Q-2Q-3Q

R

)1on width quantizati

range(log bits ofnumber 2

Q

RB

amplitude

discrete time [k]

quantized discrete-time signal

=digital signaldiscrete-time signal

][kx ][kxQ


Discrete-Time/Digital Signals 7/10 : Quantization

• B-bit quantization:

– the quantization error can only

take on values between and– hence can be considered as a random noise

signal (see below) with range – the signal-to-noise ratio (SNR) of the B-bit quantizer can

then be defined as the ratio of the signal range and the quantization noise range :

= the “6dB per bit” rule


Discrete-Time/Digital Signals 8/10 : Reconstruction

• Reconstructor = – ‘fill the gaps’ between adjacent samples– e.g. staircase reconstructor (with `hold’ circuit):

amplitude

discrete time [k]

amplitude

continuous time (t)

reconstructed

analog signal

discrete-time/digital signal

][kx )(txR



• Ideal reconstructor =– Ideal (rectangular) low-pass filter– no distortion

magnitude

frequency (f)

magnitude

frequency

• Staircase reconstructor =– sinc-like low-pass filter with sidelobes– distortion due to spurious high frequencies

magnitude

frequency (f)

)( fX D

)( fX D



• Anti-image post-filter– low-pass filter succeeds reconstructor, to remove spurious

high frequency components due to non-ideal reconstruction

• Complete scheme is…

Digital OUT

Analog IN

DSPDigital

IN

sampler quantizeranti-

aliasing prefilter

anti-image

postfilterreconstructor

Analog OUT


Discrete-Time Systems 1/15

Discrete-time (DT) system is `sampled data’ system

Input signal u[k] is a sequence of samples (=numbers) ..,u[-2],u[-1] ,u[0], u[1],u[2],…

System then produces a sequence of output samples y[k] ..,y[-2],y[-1] ,y[0], y[1],y[2],…

Example: `DSP’ block in previous slide

u[k] y[k]



Will consider linear time-invariant (LTI) DT systems

Linear : input u1[k] -> output y1[k]

input u2[k] -> output y2[k]

hence a.u1[k]+b.u2[k]-> a.y1[k]+b.y2[k]

Time-invariant (shift-invariant) input u[k] -> output y[k], hence input u[k-T] -> output y[k-T]

u[k] y[k]



Causal systems

iff for all input signals with u[k]=0,k<0 -> output y[k]=0,k<0

Impulse response

input …,0,0, 1 ,0,0,0,...-> output …,0,0, h[0] ,h[1],h[2],h[3],...

General input u[0],u[1],u[2],u[3] (cfr. linearity & shift-invariance!)

]3[

]2[

]1[

]0[

.

]2[000

]1[]2[00

]0[]1[]2[0

0]0[]1[]2[

00]0[]1[

000]0[

]5[

]4[

]3[

]2[

]1[

]0[

u

u

u

u

h

hh

hhh

hhh

hh

h

y

y

y

y

y

y

this is called a

`Toeplitz’ matrix



Convolution

]3[

]2[

]1[

]0[

.

]2[000

]1[]2[00

]0[]1[]2[0

0]0[]1[]2[

00]0[]1[

000]0[

]5[

]4[

]3[

]2[

]1[

]0[

u

u

u

u

h

hh

hhh

hhh

hh

h

y

y

y

y

y

y

u[0],u[1],u[2],u[3] y[0],y[1],...

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

][*][][.][][ kukhiuikhkyi

= `convolution sum’



Z-Transform of system h[k] and signals u[k],y[k]

i

izihzH ].[)(

]3[

]2[

]1[

]0[

.

]2[000

]1[]2[00

]0[]1[]2[0

0]0[]1[]2[

00]0[]1[

000]0[

.1

]5[

]4[

]3[

]2[

]1[

]0[

.1

3211).()(

5432154321

u

u

u

u

h

hh

hhh

hhh

hh

h

zzzzz

y

y

y

y

y

y

zzzzz

zzzzHzY

i

iziyzY ].[)(

i

iziuzU ].[)(

)().()( zUzHzY H(z) is `transfer function’



Z-Transform• easy input-output relation:

• may be viewed as `shorthand’ notation (for convolution operation/Toeplitz-vector product)

• stability =bounded input u[k] leads to bounded output y[k] --iff

--iff poles of H(z) inside the unit circle (now z=complex variable) (for causal,rational systems, see below)

)().()( zUzHzY

k

kh ][



Example-1 : `Delay operator’ Impulse response is …,0,0 ,0, 1,0,0,0,…

Transfer function is Pole at z=0

Example-2 : Delay + feedback Impulse response is …,0,0 ,0, 1,a,a^2,a^3…

Transfer function is

Pole at z=a

1)( zzH

u[k]

y[k]=u[k-1]

x

+

a

u[k]

y[k]

azza

zzH

zzHzazH

zazazazzH

1

.1)(

)(.)(

......)(

1

1

11

433221



Will consider only rational transfer functions: • In general, these represent `infinitely long impulse response’ (`IIR’)

systems• N poles (zeros of A(z)) , N zeros (zeros of B(z))• corresponds to difference equation

• Hence rational H(z) can be realized with finite number of delay elements, multipliers and adders

NN

NN

NNN

NNN

zaza

zbzbb

zazaz

bzbzb

zA

zBzH

...1

...

...

...

)(

)()(

11

110

11

110

][....]1[.][....]1[.][.][ 110 NkyakyaNkubkubkubky NN

...)().()().()().()( zUzBzYzAzUzHzY



Special case is

• N poles at the origin z=0 (hence guaranteed stability) • N zeros (zeros of B(z)) = `all zero’ filters• corresponds to difference equation

=`moving average’ (MA) filters• impulse response is

= `finite impulse response’ (`FIR’) filters

][....]1[.][.][)().()( 10 NkubkubkubkyzUzHzY N

,...0,0,0,,,...,,,0,0,0 110 NN bbbb

NNNzbzbb

z

zBzH ...

)()( 1

10


H(z) & frequency response:• given a system H(z)• given an input signal = complex sinusoid

• output signal : = `frequency response’ = complex function of radial frequency ω = H(z) evaluated on the unit circle

)sin(.)cos(

][

kjk

keku kj

)(].[][ ].[)(].[][].[ j

i

ijeihkj

ei

ikjeihi

ikuih eHkuky

)( jeH


Re

u[0]=1

u[2] u[1]

Im

Im

Re


-4 -2 0 2 40

0.5

1

-4 -2 0 2 4-5

0

5


H(z) & frequency response:• periodic : period = • for a real impulse response h[k]

Magnitude response is even function

Phase response is odd function

• example (`low pass filter’):

)( jeH

)( jeH

Nyquist frequency

,...1,1,1,1,1..., kje

2

)( jeH

(=2 samples/period)



• H(z) & Fourier transform– the frequency response of an LTI system is equal to the

Fourier transform of the continuous-time impulse sequence (see p.7) constructed with h[k] :

– similarly, the frequency spectrum of a discrete-time signal (=its z-transform evaluated at the unit circle) is equal to the Fourier transform of the continuous-time impulse sequence constructed with u[k], y[k] :

• Input/output relation: )().()( jjj eUeHeY

k s

jsD f

feHTktkhth .2 , )(...)}.(].[{)}({ FF

k s

jsD f

feUTktkutu .2 , )(...)}.(].[{)}({ FF

)(),( jj eYeU

)( jeH



• Example: All-pass filter– a (unity-gain) all-pass filter is a filter that passes all input

signal frequencies without gain or attenuation

– hence a (unity-gain) all-pass filter preserves signal energy

– an all-pass filter may have any phase response

kk

kyku22

][][

1)( jeH



• Example: Biquadratic (=2nd order) all-pass filter– it can be shown that for the unity-gain constraint to hold,

the denominator coefficients must equal the numerator coefficients in reverse order, i.e.,

– the poles and zeros are then related as follows



• PS: – So far have only considered single-input/single-output

(SISO) systems

– Similar equations for multiple-input/multiple-output (MIMO) systems

– Example : 2-inputs, 3 outputs

)().()( zUzHzY

)().()( zzz UHY

)(

)(.

)()(

)()(

)()(

)(

)(

)(

2

1

3231

2221

1211

3

2

1

zX

zX

zHzH

zHzH

zHzH

zY

zY

zY


Discrete/Fast Fourier Transform 1/5

• DFT definition:– the frequency spectrum/response of a discrete-time

signal/system x[k] is a (periodic) continuous function of the radial frequency ω

– The `Discrete Fourier Transform’ (DFT) is a discretized version of this, obtained by sampling ω at N uniformly spaced frequencies (n=0,1,..,N-1) and by truncating x[k] to N samples (k=0,1,..,N-1)

= DFT

jez

k

k

j zkxeX

].[)(

Nnn /2

kN

njN

k

N

nj

ekxeX 21

0

2

].[)(



• Inverse discrete Fourier transform (IDFT):– an -point DFT can be calculated from an -point time

sequence:

– vice versa, an -point time sequence can be calculated from an -point DFT:

= IDFT

= DFT



• DFT/IDFT in matrix form– Using shorthand notation..

– ..the DFT can be rewritten as

– an -point DFT requires complex multiplications



• DFT/IDFT in matrix form– Using shorthand notation..

– ..the IDFT can be rewritten as

– an -point IDFT requires complex multiplications



• Fast Fourier Transform (FFT) (1805/1965)

– divide-and-conquer approach: • split up N-point DFT in two N/2-point DFT’s• split up two N/2-point DFT’s in four N/4-point DFT’s• …• split up N/2 2-point DFT’s in N 1-point DFT’s• calculate N 1-point DFT’s• rebuild N/2 2-point DFT’s from N 1-point DFT’s• …• rebuild two N/2-point DFT’s from four N/4-point DFT’s• rebuild N-point DFT from two N/2-point DFT’s

– DFT complexity of multiplications is reduced to

FFT complexity of multiplications

Jam

es W

. Coo

ley

John

W.T

ukey

Car

l Frie

dric

h G

auss

(17

77-1

855


Need more?

• Introductory books– S. J. Orfanidis, “Introduction to Signal Processing”, Prentice-Hall

Signal Processing Series, 798 p., 1996– J. H. McClellan, R. W. Schafer, and M. A. Yoder, “DSP First: A

Multimedia Approach”, Prentice-Hall, 1998– P. S. R. Diniz, E. A. B. da Silva and S. L. Netto, “Digital Signal

Processing: System Analysis and Design”, Cambridge University Press, 612 p., 2002

• Online books– Smith, J.O. Mathematics of the Discrete Fourier Transform (DFT),

http://ccrma.stanford.edu/~jos/mdft/, 2003, ISBN 0-9745607-0-7. – Smith, J.O. Introduction to Digital Filters, August 2006 Edition,

http://ccrma.stanford.edu/~jos/filters06/.

lecture 2 signals & systems review marc moonen & toon van waterschoot dept. e.e./esat,...

Documents