Basics on Digital Signal Processing
Introduction
Vassilis Anastassopoulos
Electronics Laboratory, Physics Department,
University of Patras
Outline of the Course
1. Introduction (sampling – quantization)
2. Signals and Systems
3. Z-Transform
4. The Discreet and the Fast Fourier Transform
5. Linear Filter Design
6. Noise
7. Median Filters
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Analog & digital signals
Analog
Digital
Discrete function Vk of
discrete sampling
variable tk, with k =
integer: Vk = V(tk).
Sampled
Signal
0.3
0.3
0.2
0.2
Voltage [V]
Voltage [V]
Continuous function V
of continuous variable t
(time, space etc) : V(t).
0.1
0
-0.1
-0.2
0.1
0
ts ts
-0.1
-0.2
0
2
4
6
time [ms]
8
10
0
2
4
6
8
sampling time, tk [ms]
10
Uniform (periodic) sampling.
Sampling frequency fS = 1/ tS
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Analog & digital systems
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Digital vs analog processing
Digital Signal Processing (DSPing)
Advantages
Limitations
• Often easier system upgrade.
• A/D & signal processors speed:
wide-band signals still difficult to
treat (real-time systems).
• Data easily stored -memory.
• Finite word-length effect.
• More flexible.
• Better control over accuracy
requirements.
• Reproducibility.
• Linear phase
• No drift with time and
temperature
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DSPing: aim & tools
Applications
• Predicting a system’s output.
• Implementing a certain processing task.
• Studying a certain signal.
• General purpose processors (GPP), -controllers.
Hardware
Software
• Digital Signal Processors (DSP).
Fast
• Programmable logic ( PLD, FPGA ).
Faster
real-time
DSPing
• Programming languages: Pascal, C / C++ ...
• “High level” languages: Matlab, Mathcad, Mathematica…
• Dedicated tools (ex: filter design s/w packages).
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Related areas
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Applications
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Important digital signals
Unit Impulse or Unit Sample.
δ(nTs)
δ[(n-3)Τs]
The most important signal for
two reasons
nΤs past
u(nTs)
δ(n)=1 for n=0
Unit Step u(n)=1 for n0
nΤs past
δ(n)=u(n)-u(n-1)
r(nTs)
Unit Ramp r(n)=nu(n)
nΤs past
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Digital system example
General scheme
ms
V
Sometimes steps missing
ms
A
(ex: economics);
k
- D/A + filter
(ex: digital output wanted).
Antialiasing
A
k
V
V
ms
A/D
Digital
Processing
Digital
Processing
D/A
Filter
Reconstruction
ms
ANALOG
DOMAIN
Topics of this
lecture.
A/D
DIGITAL
DOMAIN
- Filter + A/D
Filter
Filter
Antialiasing
ANALOG
DOMAIN
V
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Digital system implementation
ANALOG INPUT
Antialiasing
Filter
A/D
Digital
Processing
KEY DECISION POINTS:
Analysis bandwidth, Dynamic range
• Pass / stop bands.
• Sampling rate.
• No. of bits. Parameters.
• Digital format.
1
2
3
What to use for processing?
DIGITAL OUTPUT
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AD/DA Conversion – General Scheme
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AD Conversion - Details
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Sampling
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1
Sampling
How fast must we sample a continuous
signal to preserve its info content?
Ex: train wheels in a movie.
25 frames (=samples) per second.
Train starts
wheels ‘go’ clockwise.
Train accelerates
wheels ‘go’ counter-clockwise.
Why?
Frequency misidentification due to low sampling frequency.
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Rotating Disk
How fast do we have to instantly
stare at the disk if it rotates
with frequency 0.5 Hz?
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1
The sampling theorem
A signal s(t) with maximum frequency fMAX can be
Theo* recovered if sampled at frequency f > 2 f
S
MAX .
* Multiple proposers: Whittaker(s), Nyquist, Shannon, Kotel’nikov.
Naming gets
confusing !
Nyquist frequency (rate) fN = 2 fMAX or fMAX or fS,MIN or fS,MIN/2
Example
s(t)  3  cos(50π t)  10  sin(300π t)  cos(100π t)
F1
F2
F1=25 Hz, F2 = 150 Hz, F3 = 50 Hz
Condition on fS?
F3
fS > 300 Hz
fMAX
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Sampling and Spectrum
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1
Sampling low-pass signals
Continuous spectrum
(a)
(a) Band-limited signal:
frequencies in [-B, B] (fMAX = B).
-B
0
(b)
B
f
Discrete spectrum
No aliasing
(b) Time sampling
frequency
repetition.
fS > 2 B
-B
0
B fS/2
f
Discrete spectrum
Aliasing & corruption
(c)
0
fS/2
no aliasing.
(c) fS
f
2B
aliasing !
Aliasing: signal ambiguity
in frequency domain
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1
Antialiasing filter
(a)
(a),(b) Out-of-band noise can aliase
Signal of interest
Out of band
noise
Out of band
noise
-B
0
B
(b)
f
into band of interest. Filter it before!
(c) Antialiasing filter
Passband: depends on bandwidth of
interest.
Attenuation AMIN : depends on
(c)
-B
f
0
B fS/2
• ADC resolution ( number of bits N).
AMIN, dB ~ 6.02 N + 1.76
• Out-of-band noise magnitude.
Other parameters: ripple, stopband
frequency...
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1
Under-sampling
Using spectral replications to reduce
sampling frequency fS req’ments.
Bandpass signal
centered on fC
0
f
2  fC  B
2  fC  B
 fS 
m 1
m
m
B
fC
, selected so that fS > 2B
Example
fC = 20 MHz, B = 5MHz
Without under-sampling fS > 40 MHz.
With under-sampling fS = 22.5 MHz (m=1);
= 17.5 MHz (m=2); = 11.66 MHz (m=3).
-fS
f
0
fS
2fS
fC
Advantages
 Slower ADCs / electronics
needed.
 Simpler antialiasing filters.
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Quantization and Coding
N Quantization Levels
q
Quantization Noise
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2
SNR of ideal ADC
Assumptions
 RMS input 
 (1)
SNRideal  20  log10 
 Ideal ADC: only quantisation error eq
 RMS(e q ) 


Also called SQNR
(signal-to-quantisation-noise ratio)
RMS input 
(p(e) constant, no stuck bits…)
 eq uncorrelated with signal.
 ADC performance constant in time.
T
2
V
1  VFSR

 
 sinωt  dt  FSR
T  2
2 2

0
Input(t) = ½ VFSR sin( t).
p(e)
quantisation error probability density
q/2
RMS(e q ) 

 
eq2  p eq deq 
-q/2
VFSR
q

12 2N  12
(sampling frequency fS = 2 fMAX)
1
q
q
2
q
2
eq
Error value
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2
SNR of ideal ADC - 2
SNRideal  6.02  N  1.76 [dB]
Substituting in (1) :
One additional bit
(2)
SNR increased by 6 dB
Real SNR lower because:
- Real signals have noise.
- Forcing input to full scale unwise.
- Real ADCs have additional noise (aperture jitter, non-linearities etc).
Actually (2) needs correction factor depending on ratio between sampling freq
& Nyquist freq. Processing gain due to oversampling.
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Coding - Conventional
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Coding – Flash AD
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DAC process
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Oversampling – Noise shaping
PSD
Nyquist Sampler
f
fb
fN
The oversampling process takes apart
the images of the signal band.
(a)
Oversampling OSR=4
f
fs=4fN
(b)
PSD
Signal
Quantization noise in
Nyquist converters
Quantization noise in
Oversampling converters
0
fN/2
PSD
Signal
fs/2
Quantization noise
Nyquist converters
Quantization noise
Oversampling and noise
shaping converters
Spectrum at the output of a noise
shaping quantizer loop compared to
those obtained from Nyquist and
Oversampling converters.
Quantization noise
Oversampling converters
0
FN/2
frequency
When the sampling rate increases (4
times) the quantization noise spreads
over a larger region. The quantization
noise power in the signal band is 4 times
smaller.
Fs/2
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Digital Systems
A discreet-time system is a device or algorithm
that operates on an input sequence according to
some computational procedure
It may be
•A general purpose computer
•A microprocessor
•dedicated hardware
•A combination of all these
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Linear, Time Invariant Systems
System Properties
• linear
•Time Invariant
•Stable
•Causal
N
y ( n )   ak x ( n  k )
k 0
Convolution
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Linear Systems - Convolution
5+7-1=11 terms
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Linear Systems - Convolution
5+7-1=11 terms
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General Linear Structure
M
L
k 0
k 1
y (n)   ak x(n  k )   bk y (n  k )
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Simple Examples
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Linearity – Superposition – Frequency Preservation
Principle of Superposition
x1(n)
y1(n)
H
ax1(n)+bx2(n)
ay1(n)+by2(n)
H
x2(n)
y2n)
H
Principle of Superposition  Frequency Preservation
x12(n)
x1(n)
x2
x12(n)+x22(n)+2 x1(n) x2(n)
x1(n)+x2(n)
x2
x2(n)
2
Non-linear
x22(n)
x
If y(n)=x2(n) then for x(n)=sin(nω) y(n)=sin2(nω)=0.5+0.5cos(2nω)
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The END
Have a nice Weekend
Back on Tuesday
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