Transcript Chapter2_Lect6.ppt

Chapter 2
Linear Systems
Topics:
 Review of Linear Systems
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Linear Time-Invariant Systems
Impulse Response
Transfer Functions
Distortionless Transmission
Huseyin Bilgekul
Eeng360 Communication Systems I
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
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Linear Time-Invariant Systems
• An electronic filter or system is Linear when Superposition holds,
that is when,
y(t )  l[a1 x1 (t )  a2 x2 (t )]  a1l[ x1 (t )]  a2l[ x2 (t )]
• Where y(t) is the output and x(t) = a1x1(t)+a2x2(t) is the input.
• l[.] denotes the linear (differential equation) system operator acting on [.].
y (t )  x(t )  h(t )
Y ( f )  X ( f )H ( f )
• If the system is time invariant for any delayed input x(t – t0), the
output is delayed by just the same amount y(t – t0).
• That is, the shape of the response is the same no matter when the
input is applied to the system.
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Impulse Response
x(t)
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Impulse Response
• The linear time-invariant system without delay blocks is described by a linear
ordinary differential equation with constant coefficients and may be
characterized by its impulse response h(t).
• The impulse response is the solution to the differential equation when the
forcing function is a Dirac delta function.
y(t) = h(t) when x(t) = δ(t).
• In physical networks, the impulse response has to be causal.
h(t) = 0 for t < 0
• Generally, an input waveform may be approximated by taking samples of the
input at ∆t-second intervals.
• Then using the time-invariant and superposition properties, we can obtain the
approximate output as
• This expression becomes the exact result as ∆t becomes zero. Letting n∆t = λ,
we obtain
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Transfer Function
• The output waveform for a time-invariant network can be obtained by
convolving the input waveform with the impulse response of the system.
• The impulse response can be used to characterize the response of the
system in the time domain.
• The spectrum of the output signal is obtained by taking the Fourier
transform of both sides. Using the convolution theorem,
Y ( f )  X ( f ) H ( f ) or
Y( f )
H( f ) 
X( f )
• Where H(f) = ℑ[h(t)] is transfer function or frequency response of the network.
• The impulse response and frequency response are a Fourier transform pair:
h(t )  H ( f )
• Generally, transfer function H(f) is a complex quantity and can be written in polar form.
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Transfer Function
• The |H(f)| is the Amplitude (or magnitude) Response.
• The Phase Response of the network is
•
Since h(t) is a real function of time (for real networks), it follows
1. |H(f)| is an even function of frequency and
2. θ(f) is an odd function of frequency.
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Output for Sinusoidal and periodic Inputs
Sinusoidal Input  Sinusoidal Output with magnitude and phase change
x(t )  A cos ot
 y (t )  AH ( f o ) cos ot  H ( f o )

Any Periodic Input  All the componets of the output with magnitude and phase changes
X(f ) 
n 
 c  ( f  nf
n 

n
o
)

Y( f ) 
n 
 c H (nf
n 
n
x(t )  D0   Dn cos(not  n )  y(t)  D0 H (0) 
n 1
o
) ( f  nf o )
n 
 D H (nf
n 
n
o
) cos  not  n +H (nf o ) 
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Power Transfer Function
 Derive the relationship between the power spectral density (PSD) at the input,
Px(f), and that at the output, Py(f) , of a linear time-invariant network.
Using the definition of PSD
Px ( f )  lim
T 
XT ( f )
2
T
PSD of the output is
Using transfer function
in a formal sense, we obtain
Thus, the power transfer
function of the network is
Gh ( f ) 
Py ( f )
Px ( f )
 H( f )
2
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Example 2.14 RC Low Pass Filter
x(t )  Ri (t )  y (t )
from KVL
dy
dy
RC  y (t )  x(t )
dt
dt
RC ( j 2 f )Y ( f )  Y ( f )  X ( f ) Using Fourier Transform
Y( f )
1
H( f ) 

Transfer Function of RC
X ( f ) 1  j (2 RC ) f
i (t )  C
 1 t
 e o to
1
h(t )    H ( f )    o
Impulse Response where  o  RC

t0
0
Gh ( f )  H ( f ) 
2
fo  1
2 RC
1
1   f 
 fo 
2
Power Transfer Function
(3dB Frequency), Gh ( f o )  1
2
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Example 2.14 RC Low Pass Filter
3dB Point
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Distortionless Transmission
No Distortion if y(t)=Ax(t-Td) Y(f)=AX(f)e-j2fTd
For no distortion at the output of an LTI system, two requirements must be satisfied:
1.
The amplitude response is flat.
|H(f)| = Constant = A
( No Amplitude Distortion)
2.
The phase response is a linear function of frequency.
θ(f) = <H(f) = -2πfTd
(No Phase Distortion)
Second requirement is often specified equivalently by using the time delay.
We define the time delay of the system as:
 If Td(f) is not constant, there is phase distortion,
Because the phase response θ(f), is not a linear function of frequency.
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Example 2.15 Distortion Caused By an RC filter
 For f <0.5f0, the filter will provide almost distortionless transmission.
The error in the magnitude response is less than 0.5 dB.
The error in the phase is less than 2.18 (8%).
 For f <f0,
The error in the magnitude response is less than 3 dB.
The error in the phase is less than 12.38 (27%).
 In engineering practice, this type of error is often considered to be tolerable.
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Example 2.15 Distortion Caused By a filter
 The magnitude response of the RC filter is not constant.
 Distortion is introduced to the frequencies in the pass band below frequency fo.
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Example 2.15 Distortion Caused By a filter
 The phase response of the
RC filter is not a linear
function of frequency.
 Distortion is introduced to
the frequencies in the pass
band below frequency fo.
 The time delay across the
RC filter is not constant for
all frequencies.
 Distortion is introduced to
the frequencies in the pass
band below frequency fo.
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