Signal Processing for TPCs in High Energy Physics
(Part I)
Beijing, 9-10 January 2008
Outline
• Introduction to signal processing in HEP
• Detector signal processing model
• Electronic signal processing
• Preamplifier and Shaper
• Analogue to Digital Conversion
• Digital signal processing
Luciano Musa - CERN
Beijing, January 2008
Luciano Musa
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Signal Processing in High Energy Physics
Introduction
 Signal Processing is a way of converting an obscure signal into useful
information
 Signal processing includes signal formation due to a particle passage within
a detector, signal amplification, signal shaping (filtering) and readout
 The basic goal is to extract the desired and pertinent information from the
obscuring factors (e.g. noise, pile-up)
 The two quantities of greatest importance to be extracted from detector
signals are:
• amplitude:
energy, nature of the particle, localization
• time of occurrence:
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localization, nature of the particle
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Signal Processing in High Energy Physics
Means of Detection
 Each detection method has to extract some energy from the particle to be
detected
 Nearly all detection methods (Cerenkov and Transition Radiation Detector being an
exception) make use of ionization or excitation
 Charged particles: ionization and excitation is produced directly by the interaction
of the particle electromagnetic field with the electrons of the detection medium
 A typical particle energy (today’s experiments) is of the order of few 100MeV to
GeV, while the energy loss can be below the MeV level. This is an example of a
nondestructive method for detection of charged particles.
 All neutral particles must first undergo some process that transfers all or part of
their energy to charged particles. The detection method is destructive
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Signal Processing in High Energy Physics
Detection of Ionization (1/2)

In most ionization detectors the total ionization charge is collected using an
externally applied electrical field

Sometimes an amplification process by avalanche formation in a high electrical
field is used. Examples of detectors are:
a)
Proportional chamber (MWPC, GEM, mMegas);
b)
Time Projection Chamber
c)
Liquid-argon chamber;
d)
Semiconductor detector.

All of them provide a certain amount of charge onto an output electrode

The electrode represents a certain capacitance

For signal-processing point of view these detectors are capacitive sources, i.e.
their output impedance is dominated by the capacitance
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Signal Processing in High Energy Physics
Detection of Ionization (2/2)
 This common feature of all detectors for particle physics allows a rather unified
approach to signal processing
 Despite of common features among various detectors used in high-energy physics,
great differences exist among them
• The typical charge at the detector output can differ by six orders of magnitude
• The output capacitances can differ by the same factor
• Signal dynamics
• Pulse repetition rate
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Signal Induced by a Moving Charge
Example I
Parallel Plate Ion Chamber
QA,el = -q
Anode (A)
E
d
i
QA,ion = q
Vb
  P(x)
x
V’A(P)
x
= -q
V’A
d
V’A(P)
x
V’A
=q
Applying
Green’s
Theorem
d
A constant induced current flows in the external circuit
Cathode (C)
i=
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Luciano Musa
dQA,el
dt
=-
q dx
d dt
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Signal Induced by a Moving Charge
Cylindrical Proportional Chamber
Example II
Avalanche
region
(amplification)
Charged
particle
primary
ionization
cathode
gas
anode
electron – ion
pair
Electron
cloude
E≠0
Ion
cloude
i(t) = i0 / (1+t/t0)
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Detector Signal Processing Model
series
series
white noise
1/f noise
e2W=a
e2f=c/|f|
noiseless
preamplifier
signal
A
processor
Q · s(t)
Cd
i2W=b
i2f=d·f
parallel
parallel
white noise
f noise
Ci
If s(t) = d(t)
A(Q/(Cd+Ci))
noise power spectral density
Ax[a+b/w(Cd+Ci)2]
The detector is modeled as a current source, delivering a current pulse
with time profile s(t) and charge Q, proportional to the energy released,
across the parallel combination of the detector capacitance Cd and the
preamplifier input capacitance Ci.
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Electronic Signal Processing
Signal Processor
F(f)
U(f)
h(f)
F(f)
U(f)
f
Noise floor
f0
f0
f0
f
f
Improved
Signal/Noise Ratio
Example of signal filtering - the figure shows a “typical” case of noise filtering
In particle physics, the detector signals have very often a very large frequency
spectrum
The filter (shaper) provides a limitation in bandwidth, and the output signal shape
is different with respect to the input signal shape.
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Electronic Signal Processing
F(f)
U(f)
h(f)
f(t)
u(t)
f
Noise floor
f0
f0
f
f
Improved
Signal/Noise Ratio
The output signal shape is determined, for each application, by the following
parameters:
• Input signal shape (characteristic of detector)
• Filter (amplifier-shaper) characteristic
The output signal shape is chosen such to satisfy the application requirements:
• Time measurement
• Amplitude measurement
• Pile-up reduction
• Optimized Signal-to-noise ratio
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Signal Processing for Charge Measurement
OPTIMUM PROCESSOR
signal
A
noiseless
f(t)
processor
T(s)
Cd+Ci
i(t)
n(w)
System Transfer function
AT(s) / (s  (Cd+Ci))
System Impulse response function vs(t) = ʆ-1AT(s) / (s  (Cd+Ci))
Output root mean square noise [vN2]1/2
Optimum Processor maximize
r = {Q  A/(Cd+Ci)  MAX ʆ-1(I(s)  T(s) / s) } / [vN2]1/2
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Electronic Signal Processing
R
Vout
C
Vin
Vout =
Vout =
Low-pass (RC) filter
Xc
Vin
Xc  R
Xc =
1
1
=
j 2fC jwC
1
Vin
1  RCjw
Example RC=0.5
s=jw
Integrator s-transfer function
H(s) = 1/(1+RCs)
Impulse rsponse function
2
Step function response
1 t / RC
h (t ) =
e
RC
|h(s)|
1
1
0.5
1.5
0.8
0.2
0.6
1
0.1
0.4
0.05
0.2
0.5
1
2
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4
5
t
0.01
0.05 0.1
0.5
1
Log-Log scale
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10
1
2
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f
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Electronic Signal Processing
C
Vin
Vout
R
Vout =
R
Vin
Xc  R
Vout =
RCjw
Vin
1  RCjw
Xc =
1
1
=
j 2fC jwC
High-pass (CR) filter
Example RC=0.5
s=jw
Differentiator time function
h (t ) = d (t ) 
1 t / RC
e
RC
H(s) = RCs/(1+RCs)
Step function response
|H(s)|
1
1
1
0.5
0.5
0.8
1
2
-0.5
3
4
0.6
5
0.2
0.4
-1
0.1
-1.5
0.05
0.2
-2
0.01
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0.05 0.1
0.5
1
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10
1
2
3
4
5
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Electronic Signal Processing
Combining one low-pass (RC) and one high-pass (CR) filter :
R
C
1
Vout
Vin
C
R
Vout =
RCjω
(1  RCjωC2
Vin
Example RC=0.5
s=jw
CR-RC s-transfer function
CR-RC time function
h(t) = (1  t/RC) et/RC
|h(s)|
h(s) = RCs/(1+RCs)2
Step function response
0.2
0.15
1
0.175
0.1
0.15
0.8
0.07
0.6
0.125
0.05
0.1
0.4
0.075
0.03
0.2
0.05
0.02
1
2
-0.2
3
4
5
0.025
0.015
0.01
0.05 0.1
0.5
1
Log-Log scale
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10
2
4
6
8
10
12
14
f
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Electronic Signal Processing
Combining N low-pass (RC) and one high-pass (CR) filter :
R
C
1
Vout
Vin
C
R
N times
Vout =
Example RC=0.5, n=5
s=jw
RCjω
(1  RCjωCn
Vin
CR-RC4 s-transfer function
H(s) = RCs/(1+RCs)5
CR-RC4 time function
|H(s)|
h(t) = (4  t/RC). t 3e  t/RC
Step function response
0.02
0.01
0.012
0.005
0.01
0.002
0.01
0.008
0.001
0.0075
0.005
0.0025
0.0005
0.006
0.0002
0.004
0.0001
2
4
-0.0025
6
8
0.002
10
0.001
0.0050.01
0.05 0.1
0.5
-0.005
Log-Log scale
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2
4
6
8
10
f
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Preamplifier - Shaper
I
d(t)
O
Preamplifier
Shaper
n(t)
u(t)
What are the functions of the preamplifier and the shaper (in an ideal world) ?
• Preamplifier - An ideal integrator : it detects an input charge burst Q d(t).
The output is a voltage step Q/Cf•u-1(t). It has a large signal gain such that
the noise of the subsequent stage (shaper) is negligible.
• Shaper - A filter with : characteristics fixed to give a predefined output
signal shape, and rejection of (input) noise components outside of the useful
output signal band.
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Preamplifier - Shaper
Active Integrator (“charge-sensitive amplifier”)
Inverting Voltage Amplifier
Cf
dVo / dVi = -A
Qi
Zi= 
Cd
Vi
 vo = - A vi
Input Impedance = 
V0
(no signal current flows into amplifier input)
Voltage difference across Cf:
vf = (A+1)vi
Charge deposited on Cf:
Qf = Cfvf = Cf(A+1)vi
Qi=Qf (since Zi = )
Effective Input capacitance
Gain
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Q
C = i = C (A1)
i v
f
i
dVo Avi A A 1 1
A =
=
= =
 
Q dQ Cv C A1 C C
i ii i
f f
Luciano Musa
(A>>1)
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Preamplifier - Shaper
Active Integrator (“charge-sensitive amplifier”)
Inverting Voltage Amplifier
Cf
dVo / dVi = -A
Qi
Zi= 
Cd
Vi
 vo = - A vi
Input Impedance = 
V0
(no signal current flows into amplifier input)
Qi is the charge flowing into the preamplifier ….
but some charge remains on Cdet
What fraction of the signal charge is measured?
Q
Cv
C
i=
i i = i  Qs = 1  1
C
Qs Q  Q Qs C C
det i
i det 1 det
C
i
(if Ci >> Cdet)
A=103
Cf = 1pF
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Ci = 1nF
Cdet = 10pF
Luciano Musa
Qi / Qs = 0.99
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Preamplifier - Shaper
Preamplifier
I
Shaper
O
1
1
0.8
0.8
0.6
0.6
t
0.4
0.4
t
0.2
0.2
1
2
3
4
5
-0.2
1
2
3
4
5
5
0.2
n(t)
2
d(t)
f
1
0.5
0.15
f
0.1
0.07
0.05
0.03
0.2
0.02
0.1
0.2
0.5
1
2
5
10
0.015
0.01
Ideal Integrator

Output signal for
an “ideal” input charge
0.125
1
5
10
TRANSFER FUNCTION
RCs
(1 RCs)2
0.175
0.15
0.5
CR-RC shaper
1
s
H(s) =
0.05 0.1
O(s) =
RC
(1 RCs) 2
0.1
t

t
o(t) =
e RC
RC
0.075
0.05
0.025
2
4
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8
10
12
14
t
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Preamplifier - Shaper
Preamplifier
I
Shaper
O
1
0.01
0.8
0.0075
0.6
0.005
t
0.4
t
0.0025
0.2
2
4
6
8
10
-0.0025
1
2
3
4
5
-0.005
5
0.02
2
0.005
0.01
d(t)
n(t)
f
1
0.5
ff
0.002
0.001
0.0005
0.0002
0.2
0.0001
0.1
0.2
0.5
1
2
5
10
0.001
Ideal Integrator
0.05 0.1
0.5
1
CR-RC4 shaper
1
s
H(s) =
0.0050.01

TRANSFER FUNCTION
RCs
(1 RCs)5
0.1
0.08
Output signal for
an “ideal” input charge
0.06
O(s) =
R C
(1 RCs)5
0.04
0.02
o(t) =
5
10
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20
25
30
35
t4
R C
4 t
 e R C

t
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Preamplifier - Shaper
Basic scheme of a Preamplifier-Shaper structure
Cf
n Integrators
Diff
Vout
Cd
T0
T0
T0
Semi-Gaussian Shaper
n
Vout(s) = Q/sCf . [sT0/1+ sT0].[A/(1+ sT0)]
n
n
n
Vout(t) = [QA n /Cf n!].[t/T0s] .e
-nt/Ts
Peaking time Ts = nT0
Vout(peak) = QAn nn /Cf n!en
1
0.8
0.6
0.4
0.2
2
Vout (normalized to 1 ) vs. n
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4
5
6
7
Vout peak vs. n
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Preamplifier - Shaper
O
I
d(t)
Preamplifier
Shaper
Non Ideal Integrator
1
(1 T s)
1
H(s) =
CR_RC shaper
z(t)
u(t)
RCs
(1 RCs)2

T1= 10  R  C
0.03
O(s) =
0.02
0.01
5
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RCs
(110 RCs) (1 RCs)2
-t/RC[RC  RC e9t/10RC  9 t]
o(t) = e
81 (RC)2
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Preamplifier - Shaper
O
I
d(t)
Preamplifier
Shaper
Non Ideal Integrator
1
(1 T s)
1
H(s) =
z(t)

CR_RC shaper
pz cancellation
u(t)
1 T s
1
(1 RCs)2
Pole-zero cancellation
0.175
0.15
O(s) =
0.125
0.1
RC
(1 RCs)2
t
o(t) = t e RC
RC
0.075
0.05
0.025
2
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Preamplifier - Shaper
Basic scheme of a Preamplifier-Shaper structure with p-z cancellation
Cf
Diff
Rp
N Integrators
Vout
Cd
Tp
T0
T0
Semi-Gaussian Shaper
1 sTp  A 
Q

Vout (s) =



(1 sT ) C 1 sT 1 sT 
0 
0
f f
n
By adjusting Tp such that Tp = Tf, we obtain the
same shape as with a perfect integrator
Ts = nT0
n n
Vout(t) = Q A  n  t e
C n! Ts
f
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 nt
Ts
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Real-time Digital Filters
Analogue vs. Digital Filters
Advantages of Digital Filters

Finite-duration impulse responses are achievable

Time-varying filters (even self-adaptive) are realized without any
special component by simply programming a different set of
numbers in the filter

Certain realization problems, such as negative element values, and
practical problems, such as inconveniently large components at low
frequencies, do not arise

Programmability

Greater accuracy is achieved

No sensitive to environmental conditions (e.g. temperature, supply
voltages, etc.)
Disadvantages of Digital Filters

Power

Limited by A/D speed and resolution
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Analogue to Digital Converter
Example - 4 bits A/D Converter
D0
analog
input
Analog
ADC
4-bit
Input
D1
D2
0100 1 0111…
1
Digital
Output
D3
t
Output
Code
Full scale amplitude
1111
LSB=Full scale/2N
62.5 mV for 1V/4bits Sampling
Clock
reconstructed
signal
t
16 possible output codes
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1
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0000
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Analogue to Digital Converter
Time and Amplitude Resolution
Limiting Factors in the Time and Amplitude resolution
• Time: Aperture time and Clock Jitter
• Amplitude: Noise floor
signal
jitter
clock
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Analogue to Digital Converter
Accuracy – Speed
ENOB
22
20
18
16
14
12
10
8
6
4
2
0
10K
Heisenberg
1Kohm thermal
1ps jitter
Sampling
Sample/s
100K
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1M
10M
100M
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1G
10G
100G
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The Uniform Sampling Theorem
The Uniform Sampling Theorem

Introduced by Shannon in 1948 (original idea by Nyquist in 1928)

It establishes the theoretical maximum sampling interval for complete
signal reconstruction

The theorem holds (rigorously) only for physically unrealizable bandlimited signals

Band-limited signals are a good approximation of many signal
encountered in practice
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The Uniform Sampling Theorem
Fourier Spectrum of a band-limited function f(t)
F(ω) = f  t 
|F(w)|
p(w)
(low pass filter)
 wc
wc
0
F(ω) = 0
ω  ωc
ωc = 2 πf c
w
Theorem
f(t) is uniquely determined by its values at uniform time intervals that are 1/2fc
seconds apart
fn = f







n
2f
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c







=f







nπ
ω
c







Cardinal function




sin ωc t nπ 

f (t ) =
fn
ωc t nπ
n = 
n =
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Nyquist frequency
2fc
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The z-Transform
Continuous-Time
The Laplace transform reduces constant-coefficient linear
Domain
differential equations to linear algebraic equations
Discrete-Time
The z-transform reduces constant-coefficient linear
Domain
difference equations to linear algebraic equations
Four tools for the analysis, synthesis, and understanding of linear timeinvariant electronic systems, either continuous or discrete time:
• Fourier transform
• Laplace transform
• Hilbert transform
• z-transform
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The z-Transform
The ideal sampler circuit
f(t)
F(s)
fs(t)
Fs(s)
T
f(t) is a real-life causal signal

f(t) = 0 for t < 0
fs(t)
f(t)
We consider f(t) as modulating d(t)
d(t) is a train of periodic impulse
functions of area T
...
0
T
2T
3T
4T
5T
t

f s (t) = f(t) d(t) = T  f  nT δ t  nT
n=0
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The z-Transform
The sampling process regarded as a modulation process

f s (t) = f(t) d(t) = T  f  nT δ t  nT
n=0
LAPLACE TRANSFORM
    nTs
Fs (s) =  f  nT e
n=0
Unfortunately Fs(s) contains exponentials, hence is not algebraic in s.
Z-transform
z= esT
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or
s= 1 lnz
T

Z f s (t ) = F(z) =  f  nT  z n


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n=0
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Mapping the s-Plane into the z-Plane
z= esT
Im
s = σ jω
Im
z = e jωT
jw
s-plane
z-plane
s
0
0
Re
Sampled time functions that exponentially
1
Re
Oscillating sampled time functions
decrease with increasing time
Sampled time functions that exponentially
Z=1, constant or increasing functions
increase with increasing time
depending on the multiplicity of the pole
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Frequency-Domain Characteristics
jnωs t
1 
f s (t) =   f (t )e
T n=
1 
Fs (s) =   F(s jnωs )
T n=0
|F(jw)|
Function not band-limited
0
w
EFFECT OF SAMPLING
|Fs(jw)|
ws
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Aliasing or Foldover
ws
w
35
The z-Transform
|F(jw)|
Band-limited function
wc
ws  2wc
0
wc
w
wc
ws
|Fs(jw)|
2ws
ws
wc
0
2ws
w
|Fs(jw)|
ws< 2wc
3ws
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2ws
ws wc
0
wc
Luciano Musa
ws
2ws
3ws
w
36
Difference Equation
Key element of a sampled-data system
ANALOG
antialias
WORLD
filter
S&H
A&D
input samples f(nT)
DSP
reconstr.
D/A
Readout & Recording
filter
DIGITAL FILTER
ANALOG
WORLD
output samples u(nT)
Difference Equation
The operation of the filter is described by a difference equation that relates
u(nT) as a function of the present input sample f(nT) and any number of past
input and output samples
Recursion
formula
Beijing, January 2008
u nT =
q
m
L f  nT iT    K u nT iT 
i=0 i
i =1 i

Luciano Musa
37
Difference Equation
Example of a Difference Equation Computation
First-order difference equation (m=1, q=3)
u(nT) = 3f(nT) – 2f(nT – T) + 6f(nT - 2T) + 2f(nT – 3T) – u(nT-T)
fs(t)
us(t)
5
50
4
40
Input function
3
30
2
20
1
10
Ouput function
…
1
0
1
2
3
4
t/T 
0
3
2
5
4
7
6
t/T 
-10
-20
Beijing, January 2008
Luciano Musa
38
Difference Equation
The recursiveness of the Difference Equation suggest for its implementation:
a)
program a General Purpose Computer
b)
program a Digital Signal Processor
c)
Hardwired Digital Filter
Digital Filter Simulation of a First-Order Difference Equation
Recursion
formula
u nT = K u nT T   L f(nT)
1
0
u(nT)
f(nT)
L0
+
+
Adder
L0
Multiplier
T
Delay (Register)
T
K1
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Luciano Musa
39
Digital Networks
Digital filter network values are easily obtained, often by inspection
q
Lz i
U(z) i = 0 i
H(z)=
=
i
F(z) 1 m
 Kz
i
i= 1

q
m
u(nT) =  L f(nT iT)  Ku(nT iT)
i
i
i= 0
i= 1
Difference equation
Canonic Form
System Function
Feed-forward path
L0
L1
L2
Lq
fs(t)
+
Z-1
Z-1
Z-1
Z-1
us(t)
+
-K1
m delays
-K2
-Kq
IIR FILTER
-Km
Feed-back path
Beijing, January 2008
Luciano Musa
40
Digital Networks
Digital filter network values are easily obtained, often by inspection
q
Lz i
U(z) i = 0 i
H(z)=
=
i
F(z) 1 m
 Kz
i
i= 1

q
m
u(nT) =  L f(nT iT)  Ku(nT iT)
i
i
i= 0
i= 1
Difference equation
System Function
If all Ki’s are 0
Non-recursive
L0
L1
L2
fs(t)
+
Z-1
Z-1
Z-1
q delays
Beijing, January 2008
Luciano Musa
Lq
+
us(t)
FIR FILTER
41