Lecture 4, ELG3175 : Introduction to Communication Systems © S. Loyka
Review of Linear Systems
• Linear system -> the superposition principle holds:
• Time-invariant system -> shift in time does not change
the response:
i.e., the operator L[ ] does not change in time
• Linear time-invariant (LTI) system -> both, linear and
time-invariant
Lecture 4
[ ] [ ] [ ]1 1 2 2 1 1 2 2( ) ( ) ( ) ( )a x t a x t a x t a x t+ = +L L L
[ ] [ ]1 0 1 0 0( ) ( ) ( ) ( ) ,y t x t y t t x t t t= → − = − ∀L L
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Lecture 4, ELG3175 : Introduction to Communication Systems © S. Loyka
Response of LTI System
• If the input is a Dirac delta function, the output is impulse
response:
• If the input is x(t), the output is a convolution integral:
[ ]( ) ( )h t t= δL
( ) ( ) ( ) ( )* ( )y t x h t d x t h t
∞
−∞
= τ − τ τ =∫
t∆
0τ τ
( ) ( )x h tτ − τ
0 0( ) ( )x h tτ − τ
t
0t − τ
Lecture 4
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Lecture 4, ELG3175 : Introduction to Communication Systems © S. Loyka
Transfer Function
• Convolution theorem in frequency domain:
• Transfer function (frequency response):
• Amplitude response & phase response:
( ) ( )* ( ) ( ) ( ) ( )y xy t x t h t S f S f H f= ↔ =
( )( )
( )
y
x
S fH f
S f= [ ]( ) ( ) 1h t H f↔ = L
( )( ) ( )
( ) ( )
j fH f H f e
H f f
ϕ= =
= ∠ϕ
{ }
{ }1
Im ( )( ) tan
Re ( )
H ff
H f
−
ϕ =
x(t) h(t)
y(t)
( )xS ω ( )H ω ( )yS ω
Lecture 4
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Lecture 4, ELG3175 : Introduction to Communication Systems © S. Loyka
Power Transfer Function
• Power transfer function:
• Example 2.14: RC filter
2( )( ) ( )
( )
y
x
P fG f H f
P f= =
( )2
0
1( )
1 /G f
f f=
+
0
1( )
1 /H f
jf f=
+
Lecture 4L.W
. Couch, II, D
igita
l and A
nalo
g C
om
munic
atio
n S
yste
ms, 2
001.
?=
Derive it.
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Ideal (“brick-wall”) Filters
• low-pass filter (LPF)
• high-pass filter (HPF)
• band-pass filter (BPF)
Lecture 4, ELG3175 : Introduction to Communication Systems © S. Loyka
cf f
( )H f
LPF
cf f
( )H f
HPF
lf f
( )H f
BPF
uf
f∆
19-Apr-17 5(19)
Ideal Filters: LPF
• Ideal (“brick-wall”) low-pass filter (LPF)
– Frequency response:
– impulse response:
Lecture 4, ELG3175 : Introduction to Communication Systems © S. Loyka
1, ( )
0,
c
c
f fH f
f f
≤=
>
( ) ?h t =
cf f
( )H f
LPF
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Ideal Filters: HPF, BPF
• HPF: H(f)=? h(t)=?
• BPF: H(f)=? h(t)=?
• How to build BPF using LPF and HPF?
– suggest at least 2 ways
• Practical: measured in Lab 1
Lecture 4, ELG3175 : Introduction to Communication Systems © S. Loyka19-Apr-17 7(19)
Practical Filters: Butterworth
• Frequency response:
Lecture 4, ELG3175 : Introduction to Communication Systems © S. Loyka
2
1( )
1 ( / )n
n
c
H ff f
=
+
LPFlim ( ) ( )n
n
H f H f→∞
=
0 1 2 3 4 50
0.5
1ideal LPF
n=1
n=2
n=3
n=20
Q: show that
/c
f f
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Practical Filters: Butterworth
• Implementation: n=1?
– n=3, passive
– n=2, active
Lecture 4, ELG3175 : Introduction to Communication Systems © S. Loyka
Ad
op
ted
fro
m “
Wik
ipe
dia
: B
utt
erw
ort
h filte
r”
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Lecture 4, ELG3175 : Introduction to Communication Systems © S. Loyka
???
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Lecture 4, ELG3175 : Introduction to Communication Systems © S. Loyka
Signal Bandwidth
• Defined for positive frequencies only.
• Informal: a frequency band over which a substantial (or
all) signal power is concentrated
• Absolute bandwidth: for band-limited signals, frequency
band where spectrum is not zero. For all other
frequencies, the spectrum must be zero.
• 3 dB (half-power) bandwidth: frequency band where
PSD (or ESD) is not lower than -3 dB with respect to
maximum
• Zero-crossing bandwidth: frequency band limited by 1st
zero(s) in the spectrum.
Lecture 4
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Lecture 4, ELG3175 : Introduction to Communication Systems © S. Loyka
Signal Bandwidth
f∆
f
( )x
S f
f∆ f
( )x
S f
maxS
max
2
S
f∆
f
( )x
S f
absolute bandwidth 3 dB bandwidth
zero-crossing bandwidth
Lecture 4
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Lecture 4, ELG3175 : Introduction to Communication Systems © S. Loyka
Sampling Theorem
• The sampling theorem is one of the most important results. A bridge between digital and analog worlds. DSP is based on it.
• Intuitive explanation: interpolation.
• Signal sampling:
( ) ( )
( ) ( )
( )S
n
s S
n
x t x t t nT
x nT t nT
∞
δ
=−∞
∞
=−∞
= δ − =
= δ −
∑
∑
( )x tδ
Lecture 4
A.V. Oppenheim, A.S. Willsky, Signals and Systems, 1997.
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Lecture 4, ELG3175 : Introduction to Communication Systems © S. Loyka
Sampling Theorem
• Let x(t) be bandlimited,
• Sample it at multiples of sampling interval Ts,
• x(t) is uniquely determined by its samples
• x(t) can be reconstructed as follows:
where , - sampling frequency
max( ) 0,
xS f f f= ∀ ≥
( ){ } max, 1/(2 )
s sn
x nT T f+∞
=−∞
≤
( ){ }sn
x nT+∞
=−∞
( ) ( ) ( )
( ) [ ]?
2 sinc 2
sinc /
S S S
n
fS S
n
x t f T x nT f t nT
x nT t T n
∞
=−∞
∞′=
=−∞
′ ′ = −
→ −
∑
∑
max max
1
S
f f fT
′≤ ≤ −max
12
s
S
f fT
= ≥
Lecture 4
- Nyquist frequencymax2 f
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Lecture 4, ELG3175 : Introduction to Communication Systems © S. Loyka
Sampling Theorem
( )x
S ω
( )Sδ ω
( )Sδ ω
original signal
sampled signal: fs>2fmax
sampled signal: fs<2fmax
( ) ( ) ( )S
n
x t x t t nT
∞
δ
=−∞
= δ −∑
( ) ( )* ( )
1( )
x
x s
s n
x t S S
S nT
δ δ
∞
=−∞
↔ ω ω
= ω− ω∑
Lecture 4
A.V. Oppenheim, A.S. Willsky, Signals and Systems, 1997.
Nyquist
frequency:
,min max2sf f=
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Lecture 4, ELG3175 : Introduction to Communication Systems © S. Loyka
Recovery of Sampled Signal
( )x tδ
( )x
Sδω
( )x
S ω
( )x
Sδω
( )r
xS ω
( )r
xS ω
• sample the signal with fs>2fmax
• use low-pass filter with fmax< fc< fs-fmax
to recover original signal
Lecture 4A
.V. O
ppenheim
, A.S
. Wills
ky, S
ignals
and S
yste
ms, 1
997.19-Apr-17 16(19)
Lecture 4, ELG3175 : Introduction to Communication Systems © S. Loyka
• A real signal of bandwidth B can be recovered from
samples over a time interval T provided that
Dimensionality Theorem
2N BT=
Lecture 4
1BT ≫
• This is a foundation for digital communications,
signal processing, Internet, etc.
• If not band-limited?
• Sampling in practice:
• Example: 1h of HiFi music, N=?
2sf B f= +∆
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Lecture 4, ELG3175 : Introduction to Communication Systems © S. Loyka
On Time/Bandlimited SignalsLecture 4
• An absolutely bandlimited signal cannot be
time limited and vice versa.
• Engineering implications?
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Lecture 4, ELG3175 : Introduction to Communication Systems © S. Loyka
Summary
• Review of linear systems. Response of LTI system in time &
frequency domains. Power transfer function.
• Signal bandwidth.
• Sampling theorem. Recovery of sampled signals.
• Good reference on filters (etc.): W. Siebert, Circuits, signals,
and systems, McGraw Hill, ch. 15.
• Homework: Couch, 2.6, 2.7, 2.9-2.11; Oppenheim & Willsky, Ch. 2.2,
2.3, 6.1-6.4, 7.1-7.3. Study carefully all the examples (including end-of-
chapter study-aid examples), make sure you understand and can solve
them with the book closed.
• Do some end-of-chapter problems. Students’ solution manual provides
solutions for many of them.
Lecture 4
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