1. Chapter 2 Signals & Systems Review
Marc Moonen & Toon van Waterschoot
Dept. E.E./ESAT, K.U.Leuven

2. Signals & Systems Review
Discrete-Time/Digital Signals (10 slides)
sampling, quantization, reconstruction
Discrete-Time Systems (15 slides)
LTI, impulse response, convolution, z-transform, frequency
response, frequency spectrum, IIR/FIR
Discrete/Fast Fourier Transform (5 slides)
DFT-IDFT, FFT 2

3. Introduction: Digital Signal Processing?
Signal = a physical quantity (e.g. voltage/current, intensity/grey-level,...) that varies as a function of some independent variable(s),
(e.g. time, horizontal/vertical position, …)
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’
here 3

4. Introduction: Digital Signal Processing?
Analog signal processing
Analog Domain (Continuous-Time Domain)
Jean-Baptiste Joseph Fourier ( )
Analog Signal Processing Circuit
Analog IN
Analog OUT
(=Spectrum/Fourier Transform) 4

5. Introduction: Digital Signal Processing?
Digital signal processing in an analog world
Analog domain
Digital
domain
Analog-to-
Digital
Conversion
Digital-to-
Analog
Conversion
DSP
Analog IN
Digital IN
Digital OUT
Analog OUT 5

6. Discrete-Time/Digital Signals 1/10
Analog-to-
Digital
Conversion
Digital-to-
Analog
Conversion
DSP
Analog IN
Digital IN
Digital OUT
Analog OUT
sampling
reconstruction
& quantization 6

7. Discrete-Time/Digital Signals 2/10 : Sampling
Time-domain sampling
amplitude
amplitude
discrete-time [k]
continuous-time
signal
discrete-time
signal
impulse train
continuous-time (t)
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 = 7

8. Discrete-Time/Digital Signals 3/10 : Sampling
So what does this spectrum of x_D(t) look like…
Spectrum replication
time domain:
frequency domain:
magnitude
magnitude
frequency (f)
frequency (f) 8

9. Discrete-Time/Digital Signals 4/10 : Sampling
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 9

10. Discrete-Time/Digital Signals 5/10 : Sampling
Sampling theorem
terminology:
sampling frequency/rate fs
Nyquist frequency fs/2
sampling interval/period Ts
e.g. CD audio: fs = 44,1 kHz
Harry Nyquist (1889 –1976)
Anti-aliasing prefilters
if then frequencies above the Nyquist frequency are ‘folded’ into lower frequencies (=aliasing)
to avoid aliasing, sampling is usually preceded by (analog-domain) low-pass (=anti-aliasing) filtering 10

11. Discrete-Time/Digital Signals 6/10 : Quantization
B-bit quantization
quantized discrete-time signal
=discrete-amplitude/time signal
=digital signal
discrete-time signal
amplitude
discrete time [k]
amplitude
discrete time [k] Q 2Q 3Q -Q
-2Q
-3Q R 11

12. 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 :
e.g. CD audio: 16bits -> 96dB SNR (LP’s: 60dB SNR)
6dB per bit rule 12

13. Discrete-Time/Digital Signals 8/10 : Reconstruction
Reconstructor =
‘fill the gaps’ between adjacent samples
e.g. staircase reconstructor (see also next page) :
amplitude
discrete time [k]
amplitude
continuous time (t)
discrete-time/digital signal
reconstructed
analog signal 13

14. Discrete-Time/Digital Signals 9/10 : Reconstruction
x_D(t) is generated first as an intermediate signal (after D-to-A &
sampler), which is then followed by (analog domain) filtering
Ideal reconstructor =
ideal (rectangular) low-pass filter
no distortion
magnitude
frequency (f)
magnitude
frequency (f)
Staircase reconstructor =
D-to-A followed by `sample&hold’
`hold’ corresponds to sinc-like
low-pass filter with sidelobes
distortion due to
spurious high frequencies 14

15. Discrete-Time/Digital Signals 10/10 : Reconstruction
Anti-image post-filter
low-pass (analog domain) filter succeeds reconstructor, to remove spurious high frequency components due to non-ideal reconstruction
Complete scheme is…
Digital OUT
Analog IN
DSP
Digital IN
sampler
quantizer
anti-aliasing prefilter
anti-image postfilter
reconstructor
Analog OUT 15

16. 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]

17. Discrete-Time Systems 2/15
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]

18. Discrete-Time Systems 3/15
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!)
this is called a
`Toeplitz’ matrix

19. Discrete-Time Systems 4/15
Convolution
u[0],u[1],u[2],u[3]
y[0],y[1],...
h[0],h[1],h[2],0,0,...
= `convolution sum’

20. Discrete-Time Systems 5/15
Z-Transform of system h[k] and signals u[k],y[k]
H(z) is `transfer function’

21. Discrete-Time Systems 6/15
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)

22. Discrete-Time Systems 7/15
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…
Pole at z=a
u[k]
y[k]=u[k-1] x + a
u[k]
y[k]

23. Discrete-Time Systems 8/15
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

24. Discrete-Time Systems 9/15
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

25. Discrete-Time Systems 10/15
u[0]=1
u[2]
u[1] Im
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 Re Im Re

26. Discrete-Time Systems 11/15
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’):
Nyquist frequency
(=2 samples/period)

27. Discrete-Time Systems 12/15
Z-transform & 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 (=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:
(from page 20)

28. Discrete-Time Systems 13/15
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

29. Discrete-Time Systems 14/15
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

30. Discrete-Time Systems 15/15
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

31. 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

32. Discrete/Fast Fourier Transform 2/5
DFT & Inverse DFT (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:
= DFT
= IDFT

33. Discrete/Fast Fourier Transform 3/5
DFT/IDFT in matrix form
Using shorthand notation..
..the DFT can be rewritten as
an -point DFT requires complex multiplications

34. Discrete/Fast Fourier Transform 4/5
DFT/IDFT in matrix form
Using shorthand notation..
..the IDFT can be rewritten as
an -point IDFT requires complex multiplications

35. Discrete/Fast Fourier Transform 5/5
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
James W. Cooley
Carl Friedrich Gauss (
John W.Tukey

36. Need more? Introductory books Online 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), , ISBN
Smith, J.O. Introduction to Digital Filters, August 2006 Edition,