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Communication Systems, 5e

Chapter 2: Signals and Spectra

A. Bruce CarlsonPaul B. Crilly

© 2010 The McGraw-Hill Companies

Chapter 2: Signals and Spectra

• Line spectra and fourier series• Fourier transforms• Time and frequency relations• Convolution• Impulses and transforms in the limit• Discrete Fourier Transform (new in 5th ed.)

3

Fourier Transform

• Time to frequency domain

• Frequency to time domain

• Condition …

dttf2jexptvfV

dftf2jexpfVtv

dttvE02

Table T.1 on pages 780-780

dttwjtvwV

exp

dwtwjwVtv exp21

4

Modulation (Mixing)

• Frequency translation due to real or complex mixing products

tf2jexp21tf2jexp

21tx

tf2costxtz

00

0

0

j

0

j

ff2

eff2

efXfZ

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Spectrum of Real Cosine

Spectrum of cos 2 cA f t What happens when you convolve?

• Two copies shifted in frequency by +/- fc

6

Modulation

• Multiplication in the time domain is convolution in the frequency domain

tftxtz 02cos

0

j

0

j

ff2

eff2

efXfZ

• For frequency domain analysis – convolve• For time domain analysis – use trig function identities

Table T.3 on pages 851-853

Multiplication-Convolution• Convolution in the time domain

• The Fourier Transform Pairs are:

7

dtvwdtwvtwtv

fWfVtwtv

fWfVtwtv *

fWfVdevfW

dvefWdvdtetw

dtedtwvtwtvF

fj

fjtfj

tfj

2

22

2

Table T.3

9

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Frequency Translation- Complex Mixing tfjtvts c 2exp cffjVfS *

(a) Initial Signal Spectrums (b) Output Spectrum

cff

10

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(b) Magnitude spectrum

RF Pulse Mixing tftAts c

2cos

cc ffAffAfS sinc2

sinc2

Is convolution easier?(a) RF Pulse

11

Mixing

RF Input IF Output

LocalOscillator

tLOtRFtIF

LOLOLO tfAtLO 2cos

RFRFRF tfAtRF 2cos

LOLOLORFRFRF tfAtfAtIF 2cos2cos

LOLORFRFRFLO tftfAAtIF 2cos2cos

12

Trigonometry Identities

sincoscossinsin

sincoscossinsin

sinsincoscoscos

sinsincoscoscos

cos21cos

21sinsin

sin21sin

21cossin

cos21cos

21coscos

sin21sin

21sincos

Table T.3 on pages 851-853

13

Mixing (2)

LOLORFRFRFLO tftfAAtIF 2cos2cos

LORFLORFRFLO

LORFLORFRFLO

tffAA

tffAAtIF

2cos21

2cos21

LORFLORFRFLO tffAAtIF 2cos21

• Restating

• Using an Identity

• After an Ideal Low Pass Filtering

14

Spectral Equivalent – Real Mixing

• The mixing of a real RF input with a real Cosine local oscillator– Real Signal and Cosine LO spectrum– Post mixer sum and difference spectrum– Post Low Pass Filter (LPF) result

Real Signal

Cosine

Mixing Products

LPF

Real Signal Spectrum

Mixing Cos Signal SpectrumConvolved Signal Spectrum

Low Pass Spectrum

15

Spectral Equivalent – Complex Mixing

• The mixing of a real RF input with a Complex local oscillator– Real Signal and Complex LO spectrum– Post mixer sum spectrum (convolution in freq.)– Post Low Pass Filter (LPF) result

Real Signal

Complex Oscillator

Mixing Products

LPF

Real Signal Spectrum

Mixing Exp Signal SpectrumConvolved Signal Spectrum

Low Pass Spectrum

16

Higher Order Mixing

Mixers in Microwave Systems (Part 1)Author: Bert C. Henderson WJ Tech-note

http://www.rfcafe.com/references/articles/wj-tech-notes/Mixers_in_systems_part1.pdf

Part of WJ Comm. Technical Publicationshttp://www.rfcafe.com/references/articles/wj-tech-notes/watkins_johnson_tech-notes.htm

The old Watkins Johnson and later WJ was owned by Triquint which merged with RF Micro Devices

and now is called Qorvohttps://www.qorvo.com/

https://www.qorvo.com/design-hub

DISCRETE FOURIER TRANSFORM

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Sampling

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Given a bandlimited time function ( ), the signal canbe fully represented by a periodic set of samples,

( ) ( )

where the sampling interval = , and the sampling frequencyis 1/ .

sst kT

s

s s

x t

x t x kT

t Tf T

If our sampler is a periodic impulse train, then

( ) ( ) ( ) for convenience we drop ( ) ( )

s s s

s

x kT x t t kT Tx kT x k

Sampling (2)

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x(k) is a discrete time signal and represents the samples of x(t)

Thus the samples can be stored in computer memory and processed Digital Signal Processing

( ) is an ordered sequence of numbers, possibly complexand consists of 0,1,... 1 samples.x k

k N

Sampling (3)

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For adequate representation of x(t), the sampling rate mustbe at least twice the highest frequency component in x(t)(The Nyquist Rate)

If = signal's bandwidth 2

Nyquist rate

s

s

B f B

f B

See sect. 6.1 for more information on sampling

For class applications try to use 4 to 8 times the highest frequency, they look better.

Discrete Fourier Transform Development

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12 /

0

We can insert the samples into the Fourier transform and thensubstitute the integral for a summation to get:

( ) ( ) 0,1,... 1

Discrete Fourier transform (DFT) ( ) DFT[ ( )]

Nj nk N

kX n x k e n N

X n x k

DFT (2)

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represents discrete frequency, and thus /

frequency interval = /

Like ( ), ( ) represents an ordered sequence of values which can be complex can be stored in a computer andnumerical

n s

s

n n f nf N

f f N

x n X k

ly processed

( ) ( ) ( ) ( ) ( ) ( )R I

R I

x k x k jx kX n X n jX n

Inverse DFT

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12 /

0

Inverse DFT ( ) IDFT[ ( )]

1( ) ( ) 0,1,... 1N

j nk N

n

x k X n

x k X n e n NN

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(a) Sampled rectangular pulse train

(b) N=16 point DFT

(c) analog frequency axis

kx

NknjkxnX

N

k2exp

1

0

Relationship of k= 0 to N-1 to frequency Δf = 0 to k*fs/K to fs

Discrete time monocycle and its DFT

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2

2

2*

( ) sample at 1, 100, and centered at 32

32 32( ) exp

100 100

( ) ( )

The power spectrum is ( ) ( ) ( ) ( )

ta

s

R I

uu

tx t e f aa

k kx k

X n jX n

X n X n X n P n

Discrete time monocycle and its DFT

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( )RX n

( )IX n

( )uuP n

( )x k

n

n

n

n

k(a)

(b)

(c)

(d)

(a) ( ), (b) ( ), (c) ( ), (d) ( )R I uux k X n X n P n

Convolution and system output using the DFT

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0

1 2

To get the response of a discrete time system, we do a discrete linear convolution:

( ) ( ) ( ) ( ) ( )

Where ( ) is system's impulse response,length of ( ), and ( ) are and re

N

l

y k x k h k x l h k l

h kx k h k N N

1 2

spectively, andlength of ( ) is ( 1)

( ) system function ( ) ( )

( ) ( ) ( ) ( ) IDFT[ ( )]

y k N N N

H n h k H n

Y n H n X n y k Y n

MATLAB filtfunction

Circular convolution and the DFT

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1 1 2 2

1 2 1 2

If ( ) DFT[ ( )] and ( ) DFT[ ( )]

( ) ( ) ( ) ( )

Where denotes circular convolution.

Circular convolution both functions are the same length and the resultant function's length can be

X n x k X n x k

x k x k X n X n

N

1 2

1 2

1

When 1 new terms circulate back to the beginningof the sequence

N N

N N N

When linear convolution = circular convolution

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1 2 1

1 2 1 2

1 1

If we constrain = and (2 1)

circular convolution = linear convolution

( ) ( ) ( ) ( ) ( )

( ) DFT[ ( )] DFT[ ( )]

( ) IDFT[ ( )]

N N N N

y k x k x k x k x k

Y n x k x k

y k Y n

We can use the DFT to perform linear convolution