Linear Time
Time-Invariant
Invariant (LTI) Systems
and
Convolution
(based on chapter 10)
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Linear time-invariant (LTI) system
input
x(t)
LTI
S
output
y(t)
y (t ) = Sx(t )
where S is an operator.
where is an operator.
LTI system obeys the following rules:
¾ linearityy :
S ( x1 (t ) + x2 (t )) = Sx1 (t ) + Sx2 (t )
S (α x(t )) = α Sx(t )
¾ time-shift invariance:
y (t − t ′) = Sx(t − t ′)
where t ′ time constant
where time constant
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Examples of LTI system:
y (t ) = 10 x(t )
•constant-gain system
•linear combination of time-shifts of the input signal
y (t ) = 3 x(t ) + 5 x(t − 4) − 2 x(t + 6)
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Convolution is a math operation, which takes two
functions f(t) and g(t) and produces function y(t) according to:
+∞
f (t ) ∗ g (t ) = y (t ) =
∫
f (τ )g (t − τ )dτ
−∞
Convolution
integral
where t is a parameter and τ is a variable. ¾ all LTI systems can be represented by convolution integral.
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Convolution
•convolution is a
mathematical operator
which takes two
functions f and g and
produces a third function
which represents
p
the
overlap between
f and a reversed and
translated version of gg.
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Convolution
•One function (f, for
example) is taken to be
fixed, while g is
transformed (flipped and
shifted)
•function y(t) is:
∞
y (t ) =
∫
f (τ )g (t + τ )dτ
−∞
integration range depends on the domain,
not necessarily time domain
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Some properties of convolution:
f ∗g = g∗ f
Commutativity:
f ∗ ( g ∗ h) = ( f ∗ g ) ∗ h
Associativity:
α ( f ∗ g ) = (α f ) ∗ g = f ∗ (α g )
Scalar multiplication:
Convolution theorem:
F ( f ∗ g ) = k F ( f )iF ( g )
F
- Fourier transform
where k is the normalization constant
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Impulse response
¾ if LTI input x(t) is a delta-function x(t)=d(t) (called impulse)
then output of LTI is an impulse response h(t ) function;
¾ any LTI system
t
can b
be characterized
h
t i d
by its impulse response function h ( t ) ;
¾ for any input function x(t), the output y(t) can be calculated as a
convolution of the input with the system's
system s impulse response:
y (t ) = x(t ) ∗ h(t )
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LTI system properties:
¾ an LTI system is causal if output at any time t depends only
upon
pon input;
inp t
Causality = no output until input
¾ an LTI system is memoryless if output at any time t depends only
upon input at time t ;
¾ an LTI system is stable if every input produces output.
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Cascade system
LTI -1
LTI -2
h1 (t )
h2 (t )
x(t)
y(t))
y(
Same behavior LTI
h1 (t ) ∗ h2 (t )
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C
Consider
id th
the ffollowing
ll
i
LTI system:
t
•input voltage is the sum of iR voltage
drop across resistor plus voltage
across capacitor.
S th
So,
the iinput/output
t/ t t relation
l ti is:
i
Vin (t ) = RC
dVout
+ Vout
d
dt
With the solution:
Vout (t ) = e
−t τ
t
∫V
in
(t ′)
−∞
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et ′ τ
τ
dt ′
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Suppose, input consists of
δ-functions (3 spikes)
Vin (t1 )δ (t1 − t ).....
Vout (t ) = {
0, t < t1
V (t1 )τ −1e −(t −t1 ) τ , t ≥ t1
Time t1 is usually taken as zero and v(t1)=1, then
0, t < 0
Vout (t ) = { 1
τ
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e −t τ , t ≥ 0
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•input 3 impulses different
intensity;
•output
t t is
i a linear
li
superposition
iti
of inputs;
•each response scaled and
translated accordingly;
•each
each response occurs only after
impulse which evoked it
(causality principle)
The description of output in terms of exp’s is complicated since
requires separate equations for each region
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Output is a superposition
of inputs:
3
Vin (t ) = ∑ V (ti )δ (t − ti )
i =1
3
Vout (t ) = ∑ V (ti )h(t − ti )
i 1
i=
The output is constructed from the input using transformation:
δ (t − ti ) → h(t − ti )
Means that impulse at ti evokes the corresponding response at this time.
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•the
the response from each spike
exists continuously following ti;
•the
the resultant output at t>t3
consists of contributions from
all impulses that occurred before t
For continuous signal replace sum by integral to get:
∞
Vin (t ) =
∫V
in
(ti′)δ (t − t ′)dt ′
−∞
Then, the linear superposition output is:
∞
∫V
Vout (t ) =
in
(ti′)h(t − t ′)dt ′ =
−∞
∞
t
∫V
in
(ti′)h(t − t ′)dt ′
−∞
∞
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Frequency domain description is equally valid.
1
Vin (t ) =
2π
∞
iωt
∫ Vin (ω )e dω
and
−∞
Vout (t ) =
1
2π
∞
∫V
out
(ω )eiωt dω
−∞
In frequency
q
y domain differential equation
q
is transformed into algebraic
g
equation.
q
The LTI system output in frequency domain:
Vout (ω ) =
1
Vin (ω )
1 + iωτ
Vout (ω ) = H (ω )Vin (ω )
where H(w) is a system transfer function.
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X (ω )
LTI
H (ω )
Y (ω )
¾ the
h time
i domain
d
i representation
i off output is
i function
f
i h(t),
h( )
which is a Fourier Transform of the transfer function H(ω);
¾ convolution theorem is still valid in frequency domain.
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ELECTRONIC FILTERS
(BASED ON CHAPTER 11)
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Electronic filter removes unwanted
t d noise
i componentt andd enhances
h
signal.
i l
Some examples of filters:
•passive or active
passive, because do not depend upon an external power supply ; op.amps
in active filters require the outside power supply.
•analog or digital
•linear or non-linear
linear means linear operator is applied to a time-varying signal; non-linear
means the output
p is not a linear function of its input.
p
Simplest passive linear filters are based on combinations of
resistors and capacitors (RC) or resistors and inductors (RL)
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Consider RC-filter:
LTI
X in (ω )
Transfer function: H (ω ) =
Let :
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H (ω ) =
Yout (ω )
H (ω )
Yout (ω )
,where ω = a + ib is a complex frequency.
X in (ω )
1
1 + ω 2τ 2
−i
ωτ
1 + ω 2τ 2
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Autocorrelation function of a noise signal: R (τ ) = lim
T →∞
with substitution u = t + τ ,
R (τ ) = lim
T →∞
T
∫ x (t ) x (t + τ )dτ
−T
T
∫ x (u − τ ) x (u ) du
−T
convolution of two functions x(t) and y(t)=x(-t).
The FT of R (τ ) is the power spectrum S (ω ) and
the FT of y (t )
is Y (ω ) = X * (ω ) , therefore:
S (ω ) = 2 Y (ω )Y * (ω )
(
(from
the convolution theorem and the fact that the transform of
autocorrelation function is the power spectrum)
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Consider the noise signal with power spectrum Sin (ω ) as the input
to LTI system with transfer function H (ω ) and output Sout (ω ) :
Sin (ω )
LTI
H (ω )
Sout (ω )
Using Yout (ω ) = H (ω )Yin (ω ) we can write:
Sout = 2YY * = 2 HYin H *Yin* = HH * Sin
or
Sout (ω) = H (ω)H * (ω)Sin (ω)
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The quantity controlling the noise power is H (ω )
2
- the square of the transfer function
Consider for example H(ω) in the form: H (ω ) =
1
2
1 + ω 2τ 2
when τ = 0 , H (ω ) 2 = 1
when τ
, H (ω )
2
(log scale)
¾ filters high frequencies and does not change low frequenciesthis circuit is a low-pass filter (linear
(linear-time
time invariant filter).
filter)
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Because of symmetry
y
y of S (ω ) we can consider only
y ppositive
frequencies . Then, rms amplitude is:
∞
i
2
= R (τ = 0) = ∫ S ( f )df
0
True for input
p or output:
p
∞
∞
iin
2
= Rin (τ = 0) = ∫ Sin ( f )df and iout
2
= Rout (τ = 0) = ∫ Sout ( f )df
0
0
Signal-to-noise ration (SNR):
SNRout
=
SNRin
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Rin (0)
Rout (0)
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Filter categories:
¾Low-pass filter passes low frequencies and strongly attenuates high frequencies
¾High-pass filter passes high frequencies and strongly attenuates low frequencies
¾Band-pass filter is selective; passes frequencies within the band and strongly atten
frequencies below and above the pass band.
¾ Notch filter (band-stop filter) strongly attenuates frequencies within the band
and passes frequencies below and above the pass band
band.
ω1
ω2
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Notes about SNR
SNR =
where
Psignal
Pnoise
=
I 2 signal
I 2 noise
2
P = I 2 (t ) R = I RMS
R
is the power.
db version
i off SNR
SNR:
SNR = 10 log10
I 2 signal
I 2 noise
= 20 log10
I signal
I noise
Intensity of noise grows as the bandwidth increases, while intensity of the signa
stays the same once the bandwidth covers the signal. It is important to cover on
min necessary frequency band. Band-pass filter does this job.
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