Lec 4 ELG3175sloyka/elg3175/Lec_4_ELG3175.pdfLecture 4 , ELG3175 : Introduction to Communication Systems© S. Loyka Summary • Review of linear systems. Response of LTI system in

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

19-Apr-17 1(19)


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

19-Apr-17 2(19)


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

19-Apr-17 3(19)


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.

19-Apr-17 4(19)

Ideal (“brick-wall”) Filters

• low-pass filter (LPF)

• high-pass filter (HPF)

• band-pass filter (BPF)


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:


1, ( )

0,

c

c

f fH f

f f

≤=

>

( ) ?h t =

cf f

( )H f

LPF

19-Apr-17 6(19)

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:


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

19-Apr-17 8(19)

Practical Filters: Butterworth

• Implementation: n=1?

– n=3, passive

– n=2, active


Ad

op

ted

fro

m “

Wik

ipe

dia

: B

utt

erw

ort

h filte

r”

19-Apr-17 9(19)


???

19-Apr-17 10(19)


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

19-Apr-17 11(19)


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

19-Apr-17 12(19)


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.

19-Apr-17 13(19)


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

19-Apr-17 14(19)


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=

19-Apr-17 15(19)


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)


• 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= +∆

19-Apr-17 17(19)


On Time/Bandlimited SignalsLecture 4

• An absolutely bandlimited signal cannot be

time limited and vice versa.

• Engineering implications?

19-Apr-17 18(19)


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

19-Apr-17 19(19)

Documents

Lec 4 ELG3175sloyka/elg3175/Lec_4_ELG3175.pdfLecture 4 , ELG3175 : Introduction to Communication Systems© S. Loyka Summary • Review of linear systems. Response of LTI system in