Basics on Digital Signal Processing Introduction
Vassilis Anastassopoulos Electronics Laboratory, Physics Department, University of Patras
Outline of the Course 1. Intr tro oducti tio on (s (samplin ing g – quantization) 2. Signals and Sy Systems 3. Z-Transform 4. Th The e Dis Discr cree eett and and th the e Fas Fastt Fou Fouri rier er Tr Tran ansf sfor orm m 5. Linear Fi Filter De Design 6. Noise 7. Median Filters
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Analog & digital signals Analog
Digital
Continuous function V function V of continuous continuous variable variable t (time, space etc) : V(t).
Discrete function V function Vk of discrete sampling variable tk, with k = integer: Vk = V(tk).
Sampled Signal
0.3
0.3
] V [ e 0.1 g a t 0 l o V
0.2
0.2 ] V [ e 0.1 g a t 0 l o V
-0.2
-0.2
-0.1
ts ts
-0.1
0
2
4 6 time [ms]
8
10
0
6 8 2 4 sampling time, tk [ms]
10
Uniform (periodic) sampling. Sampling frequency fS = 1/ tS 3/36
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 • Predicting a system’s output.
Applications
• Implementing a certain processing task. • Studying a certain signal.
• General purpose processors (GPP), -controllers.
Hardware
• Digital Signal Processors (DSP).
Fast
• Programmable logic ( PLD, FPGA ).
Faster
real-time DSPing
• Programming languages: Pascal, C / C++ ...
Software
• “High level” languages: Matlab, Mathcad, Mathematica… • Dedicated tools (ex: filter design s/w packages). 6/36
Related areas
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Applications
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Important digital signals Unit Impulse or Unit Sample. δ(nTs)
δ[(n-3)Τ s]
The most important signal for two reasons nΤs past
u(nTs)
δ(n)=1 for n=0
Unit Step u(n)=1 for n0 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 V
General scheme
ms V
Sometimes steps missing - Filter + A/D
Antialiasing ms
A
(ex: economics); k
- D/A + filter (ex: digital output wanted).
A
k V
Topics of this lecture.
Filter Filter Antialiasing
ms V
A/D
A/D Digital Processing
Digital Processing D/A
Filter Reconstruction
ms
D A O N M A A L I O N G D D O I M G I A T I A N L D A O N M A A L I O N G 10/36
Digital system implementation ANALOG INPUT
Antialiasing Filter
A/D
KEY DECISION POINTS: Analysis bandwidth, Dynamic range • Pass / stop bands.
1 • Sampling rate. • No. of bits. Parameters.
Digital Processing
• Digital format.
2 3
What to use for processing?
DIGITAL OUTPUT 11/36
AD/DA Conversion – General Scheme
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AD Conversion - Details
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Sampling
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Sampling
1
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 f MAX can be Theo* recovered if sampled at frequency f > 2 f . S MAX *
Multiple proposers: Whittaker(s), Nyquist, Shannon, Kotel’nikov.
Naming gets Nyquist frequency (rate) f N = 2 f MAX or f MAX or f S,MIN or f S,MIN/2 confusing ! 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 f S?
F3 f S > 300 Hz
f MAX 17/36
Sampling and Spectrum
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1
Sampling low-pass signals Continuous spectrum
(a)
(a)
Band-limited signal: frequencies in [-B, B] (f MAX = B).
-B
0
(b)
B
f
Discrete spectrum No aliasing
(b)
Time sampling repetition. f S > 2 B
-B
0
B f S/2
0
f S/2
no aliasing.
f
Discrete spectrum Aliasing & corruption
(c)
frequency
(c) f
f S
2B
aliasing !
Aliasing: signal ambiguity in frequency domain 19/36
Antialiasing filter
1 (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
0
B f S/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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Under-sampling
1
Using spectral replications to reduce sampling frequency fS req’ments. 2 f C B m 1
m
f S
Bandpass signal centered on fC
0
2 f C B
B
f C
m
, selected so that f S > 2B -f S
0
f S
Example
Slower ADCs / electronics needed.
Simpler antialiasing filters.
With under-sampling f S = 22.5 MHz (m=1);
= 17.5 MHz (m=2); = 11.66 MHz (m=3).
f C
Advantages
f C = 20 MHz, B = 5MHz Without under-sampling f S > 40 MHz.
2f S
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Quantization and Coding N Quantization Levels
q
Quantization Noise
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SNR of ideal ADC
2
RMS input (1) SNRideal 20 log10 RMS(e q )
Also called SQNR (signal-to-quantisation-noise ratio)
Assumptions Ideal
(p(e) constant, no stuck bits…)
T
ADC: only quantisation error eq
eq uncorrelated with signal. ADC performance constant in time.
2
V V RMS input FSR sinωt dt FSR T 2 2 2 1
Input(t) = ½ V FSR sin( t).
0
p( e) quantisation error probability density q/2
RMS(e q )
eq2 p eq deq - q/2
q 12
VFSR 2N 12
(sampling frequency fS = 2 fMAX)
1 q
q 2
q 2
eq Error value 23/36
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 f b
f N
The oversampling process takes apart the images of the signal band.
(a) Oversampling OSR=4
f
f s=4f N
(b) PSD Signal
Quantization noise in Nyquist converters Quantization noise in Oversampling converters
0
f N /2
PSD Signal
f s /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
F N/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
y (n )
N
a x(n k ) 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
y (n )
M
L
a x(n k ) b y(n k ) k
k 0
k
k 1
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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)+by 2(n) H
x2(n)
y2n) H
Principle of Superposition x1(n)
x2
x2
Preservation
x12(n) x1(n)+x2(n)
x2(n)
Frequency
x12(n)+x22(n)+2 x1(n) x2(n) x2
Non-linear
x22(n)
If y(n)=x2(n) then for x(n)=sin(nω) y(n)=sin2(nω)=0.5+0.5cos(2nω) 35/36
