1 basics dsp av intro
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Basics on Digital Signal Processing
Introduction
Vassilis Anastassopoulos
Electronics Laboratory, Physics Department,
University of Patras
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Outline of the Course
1. Introduction (samplingquantization)
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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-0.2
-0.1
0
0.1
0.20.3
0 2 4 6 8 10sampling time, tk [ms]
Voltage[V]
ts
-0.2
-0.1
0
0.1
0.20.3
0 2 4 6 8 10sampling time, tk [ms]
Voltage[V]
ts
Analog & digital signals
Continuous functionV
of continuousvariable t
(time, space etc) : V(t).
Analog
Discrete functionVkofdiscretesampling
variable tk, with k =
integer: Vk = V(tk).
Digital
-0.2
-0.1
0
0.1
0.20.3
0 2 4 6 8 10
time [ms]
Voltage[V]
Uniform (periodic) sampling.Sampling frequency fS= 1/ tS
SampledSignal
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Analog & digital systems
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Digital vsanalog processingDigital Signal Processing (DSPing)
More flexible.
Often easier system upgrade.
Data easily stored -memory.
Better control over accuracy
requirements.
Reproducibility.
Linear phase
No drift with time and
temperature
Advantages
A/D & signal processors speed:
wide-band signals still difficult to
treat (real-time systems).
Finite word-length effect.
Limitations
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DSPing: aim & tools
Software Programming languages: Pascal, C / C++ ...
High level languages: Matlab, Mathcad, Mathematica
Dedicated tools (ex: filter design s/w packages).
Applications Predicting a systems output.
Implementing a certain processing task.
Studying a certain signal.
General purpose processors (GPP), -controllers.
Digital Signal Processors (DSP).
Programmable logic ( PLD, FPGA ).
Hardware real-timeDSPing
Fast
Faster
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Related areas
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Applications
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Important digital signals
Unit Impulse or Unit Sample.
The most important signal for
two reasons
(n)=1 for n=0
Unit Step u(n)=1 for n0(n)=u(n)-u(n-1)
Unit Ramp r(n)=nu(n)
(nTs) [(n-3)s]
ns past
u(nTs)
ns past
r(nTs)
ns past
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Digital system example
ms
V ANALOG
DOMAIN
ms
VFilter
Antialiasing
k
A D
IGITAL
D
OMAIN
A/D
k
A
DigitalProcessing
ms
V ANALOG
DOMAIN
D/A
ms
V FilterReconstruction
Sometimes steps missing
- Filter + A/D
(ex: economics);- D/A + filter
(ex: digital output wanted).
General scheme
Topics of thislecture.
DigitalProcessing
FilterAntialiasing
A/D
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Digital system implementation
Sampling rate.
Pass / stop bands.
KEY DECISION POINTS:
Analysis bandwidth, Dynamic range
No. of bits. Parameters.
1
2
3
Digital
Processing
A/D
AntialiasingFilter
ANALOG INPUT
DIGITAL OUTPUT
Digital format.What to use for processing?
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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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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.
Frequency misidentification due to low sampling frequency.
Train starts wheels go clockwise.
Train accelerates wheels go counter-clockwise.
1
Why?
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Rotating Disk
How fast do we have to instantlystare at the disk if it rotateswith frequency 0.5 Hz?
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The sampling theorem
A signal s(t) with maximum frequency fMAX
can berecovered if sampled at frequency fS> 2 fMAX.
Condition on fS?
fS> 300 Hz
t)cos(100t)sin(30010t)cos(503s(t)
F1=25 Hz, F2= 150 Hz, F3= 50 Hz
F1 F2 F3
fMAX
Example
1
Theo*
*Multiple proposers: Whittaker(s), Nyquist, Shannon, Kotelnikov.
Nyquist frequency (rate) fN= 2 fMAXor fMAXor fS,MINor fS,MIN/2
Naming gets
confusing !
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Sampling and Spectrum
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Sampling low-pass signals
-B 0 B f
Continuous spectrum(a) Band-limited signal:
frequencies in [-B, B] (fMAX= B).
(a)
-B 0 B fS/2 f
Discrete spectrum
No aliasing (b) Time sampling frequency
repetition.fS> 2 B no aliasing.
(b)
1
0 fS/2 f
Discrete spectrum
Aliasing & corruption(c)
(c) fS 2 B aliasing !
Aliasing: signal ambiguityin frequency domain
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Antialiasing filter
-B 0 B f
Signal of interest
Out of bandnoise Out of bandnoise
-B 0 B fS/2
(a),(b)Out-of-bandnoise can aliase
into band of interest. Filter it before!
(a)
(b)
(c)
Passband: depends on bandwidth of
interest.
Attenuation AMIN: depends on
ADC resolution ( number of bits N).
AMIN, dB~ 6.02 N + 1.76
Out-of-band noise magnitude.
Other parameters: ripple, stopband
frequency...
(c)Antialiasing filter
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Under-sampling1
Using spectral replications to reduce
sampling frequency fSreqments.
m
BCf2Sf
1m
BCf2
m , selected so that fS> 2B
B
0 fC
Bandpass signal
centered on fC
-fS 0 f
S 2f
S
fC
Advantages
Slower ADCs / electronics
needed.
Simpler antialiasing filters.
fC = 20 MHz, B = 5MHz
Without under-samplingfS> 40 MHz.
With under-samplingfS= 22.5 MHz (m=1);
= 17.5 MHz (m=2); = 11.66 MHz (m=3).
Example
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Quantization and Coding
q
N Quantization Levels
Quantization Noise
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SNR of ideal ADC2
)qRMS(e
inputRMS
10log20idealSNR(1)
Also called SQNR
(signal-to-quantisation-noise ratio)
Ideal ADC: only quantisation erroreq(p(e)constant, no stuck bits)
equncorrelated with signal.
ADC performance constant in time.
Assumptions
22
FSRVT
0
dt2
tsin2
FSRV
T
1inputRMS
Input(t) = VFSRsin(t).
12N2
FSRV
12
qq/2
q/2-
qdeqep2qe)qRMS(e
eeqqError value
pp((ee))quantisation error probability density
1q
q2
q2
(sampling frequency fS= 2 fMAX)
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SNR of ideal ADC - 2
[dB]1.76N6.02SNRideal (2)Substituting in (1) :
One additional bit SNR increased by 6 dB
2
Actually(2) needs correction factor depending on ratio between sampling freq
& Nyquist freq. Processing gain due to oversampling.
- Real signals have noise.
- Forcing input to full scale unwise.
- Real ADCs have additional noise (aperture jitter, non-linearities etc).
Real SNR lower because:
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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
fs=4fN(b)
(a)
f
ffN
Oversampling OSR=4
Nyquist SamplerPSD
fb The oversampling process takes apart
the images of the signal band.
PSD
fN/20
SignalQuantization noise in
Nyquist converters
fs/2
Quantization noise in
Oversampling converters
When the sampling rate increases (4times) the quantization noise spreads
over a larger region. The quantization
noise power in the signal band is 4 times
smaller.
frequency
PSD
FN/20
SignalQuantization noise
Nyquist converters
Fs/2
Quantization noise
Oversampling converters
Quantization noise
Oversampling and noiseshaping converters
Spectrum at the output of a noiseshaping quantizer loop compared to
those obtained from Nyquist and
Oversampling converters.
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A discreet-time system is a device or algorithmthat operates on an input sequence according tosome computational procedure
Digital Systems
It may be
A general purpose computerA microprocessordedicated hardwareA combination of all these
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Linear, Time Invariant Systems
N
k
k knxany0
)()(
System Properties
linearTime InvariantStableCausal
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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L
k
k
M
k
k knybknxany10
)()()(
General Linear Structure
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Simple Examples
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LinearitySuperposition Frequency Preservation
Principle of Superposition
H
y1(n)x1(n)
H
y2n)x2(n)
H
ay1(n)+by2(n)ax1(n)+bx2(n)
Principle of Superposition Frequency Preservation
x2
x12(n)x1(n)
x2(n)
x1(n)+x2(n)
x2
x2
x22(n)
x12(n)+x2
2(n)+2 x1(n) x2(n)
If y(n)=x2(n) then for x(n)=sin(n)y(n)=sin2(n)=0.5+0.5cos(2n)
Non-linear
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The END
Back on Tuesday
Have a nice Weekend