Post on 11-May-2022
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Communication Systems, 5e
Chapter 2: Signals and Spectra
A. Bruce Carlson
Paul 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.)
What kind of math do we do?• Transmit
• Receive
3
𝑠 𝑡 = 𝐴 ⋅ 1 + 𝜇 ⋅ 𝑚 𝑡 ⋅ cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡 + 𝜃 + 𝜑 ⋅ 𝑚 𝑡 + 2𝜋 ⋅ Δ ⋅ 𝑚 𝜆 ⋅ 𝑑𝜆
𝑥 𝑡 = 𝐴 ⋅ 𝑠 𝑡 + 𝜏 + 𝑛 𝑡 ∗ ℎ 𝑡 ⋅ cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡 + 𝜗 ∗ ℎ 𝑡
𝑥 𝑡 = 𝑠 𝑡 + 𝜏 + 𝑛 𝑡There is a lot of mixing/multiplying by sin/cos and there will be convolutions (filtering), integration and decimation!
Conceptual Domains• Time Domain (Signals)
– time based waveforms
– some consistent elements
– some “random” elements (time varying, but in an understandable way – like a message)
• Frequency Domain (Spectra)– long term and short term frequency content
– consistent – Fourier transforms (Bode Plots)
– random/changing in time – Power Spectral Densities
4
Phasors and line spectra
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0 0
0 0
0 0
0
( ) cos( ) cos(2 )
where 2 frequency in radians/second 1/ period in seconds cyclical frequency in Hz phase in radians
v t A t A f t
fT ff
jtfjAtv 02expRe
Phasor representation
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(a) Phasor diagram, (b) spectrum Caution: frequencies can be positive or negative – they “rotate” in opposite directions
Euler’s theorem
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0 0( )0
cos sin
where 1
( ) cos( ) Re[ ] Re[ ]
phasor representation
j
j t j tj
e j
j
v t A t A e Ae e
Two sided spectrum
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It is often more useful to describe a signal using its two sided spectrum
0 02 20cos(2 )
2 2 pair of conjugate phasors
j f t j f tj jA AA f t e e e e
This is similar to the Fourier Transform real and imaginary frequency spectrum plot.
Usually plot “power spectrum” in dB, not amplitude or magnitude
9
Complex Phasor Notation• Complex Signal Representation (can be easier)
• Math: Euler’s theorem and related equalities
tf2cosAtv 0
exp 𝑗 ⋅ 2𝜋 ⋅ 𝑓 ⋅ 𝑡 = cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡 + 𝑗 ⋅ sin 2𝜋 ⋅ 𝑓 ⋅ 𝑡
jtf2jexpARetv 0
10
Two Dimensional Visualization• Real-Imaginary Plot of Phasor Rotation
– All real signals can be composed as the real part of a complex signal
– The real axis component will be the real output signal
tftfjtfj 000 2cos22exp2exp
11
Textbook Convention(not required for Dr. Bazuin and your solutions)
• Positive cosine representations– Convert sin to cosine
– Convert negative to 180° rotation
– The complex phasors are identical; therefore, equal
1802cos2cos 00 tfAtfA
90tf2cosAtf2sinA 00
1exp j ij
2exp
12
Classification of signals …• Periodic and non-periodic signals
• Analog and discrete signals
A discrete signal
Analog signals
A non-periodic signalA periodic signal
13
Classification of signals ..• Energy and power signals
– A signal is an energy signal if, and only if, it has nonzero but finite energy for all time:
– A signal is a power signal if, and only if, it has finite but nonzero power for all time:
• General rule: – Periodic and random signals are power signals. – Signals that are both deterministic and non-periodic are energy
signals.
𝐸 = lim→
𝑥 𝑡 ⋅ 𝑑𝑡 ⇒ 𝑥 𝑡 ⋅ 𝑑𝑡 xE0
𝑃 = lim→
1
𝑇⋅ 𝑥 𝑡 ⋅ 𝑑𝑡 xP0
14
Fourier and Laplace
• f vs. s
• The Laplace Transform (typically) deals with single sided time signals (causal)
• The Fourier Transform is one particular special case of the Laplace Transform (and usually uses two-sided time)
– See Table T.1 on p. 847-848 for Fourier Transform pairs
fjwjs 2
0
dtetfsF ts
dtetffF tfj 2
Table T.1
16
Fourier Transform• Time to frequency domain
• Frequency to time domain
• Condition … finite energy (usually assumed)(finite power signals use “Fourier Series”)
dttf2jexptvfV
dftf2jexpfVtv
dttvE02
FT – Rectangular Pulse
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2
2
2
22
2exp2exp
fj
tfjAdttfjAfV
fj
fjA
fj
fjA
fj
fjAfV
222sin2
222exp
222exp
fAf
fAfV sinc
sin
t
rectAtv
18
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Rectangular pulse spectrum
V(ƒ) = A sinc ƒv(t) = A rect t/
Note 1: I hate this phase plot, but it is required when the absolute value is used.Note 2: Is there really a difference between +180 degrees and -180 degrees on a circle?But … the frequency domain must be conjugate symmetric!
19
Periodic Signals• Relationship
– for m an integer and To the period
• Average/mean (computed from one period)
• Average Power (computed from one period)
0Tmtvtv
01
10
1Tt
t
dttvT
tv
01
1
2
0
2 1Tt
t
dttvT
tvP
20
Fourier Series (Periodic Signals with Finite Power)
• Fourier Series Coefficients
• Signal Representation
dttfn2jexptvT
1c 0
T
0n
0
n
0n tfn2jexpctv
00 T
1fwhere
n
n fnfcfV 0
21
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FS - Rectangular pulse train
dttfnjtvT
c
T
Tn
0
2
20
2exp1
0
0
2
2
00
2exp dttfnjT
Acn
The previous math at f0
00
1
Tf
FS - Rectangular pulse train (2)• Continuing
22
2
2
00
2exp dttfnjT
Acn
00
sinc fnT
Acn
Table T.2
Fourier Series Table
24
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Spectrum of rectangular pulse train
with ƒ0 = 1/4 (a) Amplitude (b) Phase
00 T
1fwhere
Note: Envelope is defined from 1 period, impulse spacing defined by periodicity
Fourier Series:The Fourier Transform of Periodic Signals
• The convolution of one period with a comb function
25
t
rectAtv fAfV sinc
n T
nf
TTfcomb 1
k
TktrectAtv
n
fT
nf
T
AfV
sinc
Time Domain Frequency Domain
k
TktT
tcomb
Fourier Transforms and Series• For a periodic signal:
– Take the Fourier Transform of one period of the waveform and plot the spectrum
• This is the “envelope” of the Fourier Series spectrum
– The Fourier Series spectrum is then a “line spectrum” with coefficients spaced at n x f0
26
Fourier Transforms and Series
27
Fourier Transform of one rect pulse
Fourier Series of an infinite rect pulse train
Fourier Series to Time Signal• If you know the Fourier Transform or Fourier Series, you
can find the time waveform … approximately.
– Inverse Fourier Transforms and series do not correctly represent discontinuities. If the original time domain curve makes a discrete change,
• The inverse Fourier Curve gives the mid-point between the two curves
• The “inverse Fourier Series” has the Gibbs Phenomenon
– Infinite energy and infinite power limitations means that the functions are approximated, but not directly solved.
28
29
Fourier-series reconstruction of a rectangular pulse train
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N
Nn0n tf2jexpctv
Matlab Gibbs Phenomenon
30
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Fourier-series reconstruction of a rectangular pulse train
See MATLAB example
31
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0dttf2jexpctvlimte0T
2N
Nn0n
N
Gibbs phenomenon at a step discontinuity
32
Useful Signal Property Definitions: Symmetry
• Symmetry (even)
• Anti-Symmetry (odd)
• Conjugate Symmetry (real is even, imag is odd)
tvtv
tvtv
*tvtv
Properties allow you to use “math tricks”to simplify computations, if you recognize them
33
Symmetry Examples (2)
• Symmetry (even)
• Anti-Symmetry (odd)
• Conjugate Symmetry (real is even, imag is odd)
tftf 2cos2cos
tftf 2sin2sin
tfconjtfj 2exp2exp
Real Signal Transformation• The Fourier Transform becomes
– The real part of V(f) has even symmetric
– The imag. Part of V(f) has odd symmetric
– For Imaginary Signals (property)• Real part of V(f) has odd symmetry
• Imag. part of V(f) has even symmetry
34
dttftvjdttftv
dtetvfV tfj
2sin2cos
2
Symmetry in Transforms• In working with transforms, there are many
properties of time signals (and spectra) that can simplify taking forward and inverse transforms– Purely real signals
– Purely imaginary signals
– Even symmetric signals
– Odd symmetric signals
– Conjugate symmetric signals
35
The properties can be used to check your results!If the result doesn’t have the property, you did the math wrong!
Even Symmetric Real Signals• The Fourier Transform becomes
36
00
22
0
'2
0
2
0'2
0
2
02
0
22
2cos2
''
''
dttftvdteetv
dtetvdtetv
dtetvdtetv
dtetvdtetvdtetvfV
tfjtfj
tfjtfj
tfjtfj
tfjtfjtfj
Conjugate Symmetric Signals• The Fourier Transform becomes
37
00
0
*
0
*
0
'2*
0
2
0'2
0
2
02
0
22
2sinIm22cosRe2
2sin2cos
''
''
dttftvjdttftv
dttftvtvjdttftvtv
dtetvdtetv
dtetvdtetv
dtetvdtetvdtetvfV
tfjtfj
tfjtfj
tfjtfjtfj
Note that the result is purely real!
38
Fourier Transform Properties• Linearity• Superposition• Time Shifting• Scale Change• Conjugation• Duality• Frequency Translation• Convolution• Multiplication• Modulation
Table T.1 on pages 847-848
Section 2.3, pp. 54-62
Section 2.4, pp. 62-68
Table T.1
Note on Fourier Transforms• Find a table
• Learn to use the table
• Yes, Calculators can do it too …but after a while you should “Know” the easy or continually repeated transformations– RectSync
– Sin/Cosdelta functions at +/- f
– Exp(i*2*pi*f*t)single delta function at f
– Convolution Multiplication
40
41
Rayleigh’s Energy Theorem• Energy in the time domain is equal to energy in
the frequency domain! (E↔v2)
dttvtvdttvE *2
dffVdtetvdttvdfefVE tfjtfj 2**2
dffVfVdffVdtetvconjE tfj *2
dffVdttvE22
42
Important Signal Property: Causality
• Signals that haven’t happened yet are not known!
• Usual application– For a single signal analysis
• Signals start at time t=0, and v(t)=0 for t<0
• Laplace transform signals
• Filters are typically defined as starting at t=0
or
– For signal processing• The signal exist up to a time t0=0, and
• v(t)=0 for t > t0
• We don’t know what comes next … but we know the history!
43
Causality and Filtering
• Convolution Form
– The filter, h, can be defined for positive time only
– The signal, x, is defined for all past time up to time t
• Then, when limited by the filter impulse response:
dtxhtz
T
0
dtxhtz
44
The RC Filter: 1st order Butterworth Low Pass Filter
• Using Rayleigh’s Energy
y(t) v(t)
RC1sRC
1
sC1R
sC1
sHsY
sV
0t,RCtexp
RC
1th
dttvdffVE22
45
Homework 2.2-8
• Percent of total Energy based on the bandwidth of an “exponential” time signal impulse response at – WBW= b/2π and WBW = 4b/2π = 2b/π
0t,0
0t,tbexpAtv
b
A
b
tbAdttbAEE tv
22
2expexp
2
0
2
0
2
fjb
AfV
2
• Total Energy is:
The exponential time signal
• Time Response
• Frequency Response
46
0t,0
0t,tbexpAtv
fjb
AfV
2
b
AEE tv
2
2
WBW=b/2π
47
Homework 2.2-8 (cont)• Percent of total Energy at various “Bandwidths” WBW
– WBW= b/2π and WBW = 4b/2π = 2b/π
WW
W
fV dfbf
Adf
bfj
A
bfj
AE
022
2
22
22
48
Homework 2.2-8 (cont)
• Percent of total Energy at– WBW= b/2π and WBW = 4b/2π = 2b/π
• For a simple RC filter – wco is the 50% power point (w in radians/sec)
RCwb co1 RCA 1
49
Realizable Filters, RC Network
Notes and figures are based on or taken from materials in the course textbook: Bernard Sklar, Digital Communications, Fundamentals and Applications,
Prentice Hall PTR, Second Edition, 2001.
The Use of Percent Total Energy
• If you want to receive a finite time signal (finite energy) signal, what bandwidth “perfect” filter should you use?– For a decaying exponential signal
• 50% of the energy received at ffilter=fco1/RC
• 84.4% received at ffilter=4 x fco 4/RC
• 93.7% received at ffilter=10 x fco 10/RC
• We usually want 90%-99% of a “pulses” energy
• This also has implication for digital sampling rates!
50
End Lecture
Another Example
• What bandwidth “ideal” filter should be use if we want to filter a bipolar square wave and receive 90% of the power?
• Use the approach just shown …– Percent energy in frequency for one period of the
periodic square wave
– The general result is the integral of the sinc^2 function from f=0 to f=?
51
52
Modulation
• Frequency translation due to real or complex mixing products (multiply in time domain)
jtfjjtfjtx
tftxtz
00
0
2exp2
12exp
2
1
2cos
0
j
0
j
ff2
eff
2
efXfZ
• Using trig functions, try cosine x cosine mixingTable T.3 on pages 851-853
Convolution in freq. domain
Table T.3
54
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Frequency translation of a bandlimited spectrum
tfjtvts c 2exp cffVfS
Baseband Complex Modulated
55
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(a) RF pulse (b) Amplitude spectrum
RF Pulse Mixing tf
tAts c
2cos
cc ffA
ffA
fS sinc2
sinc2
56
Mixing
RF Input IF Output
LocalOscillator
tLOtRFtIF
LOLOLO tfAtLO 2cos
RFRFRF tfAtRF 2cos
LOLOLORFRFRF tfAtfAtIF 2cos2cos
LOLORFRFRFLO tftfAAtIF 2cos2cos
57
Trigonometry Identities
sincoscossinsin
sincoscossinsin
sinsincoscoscos
sinsincoscoscos
cos2
1cos
2
1sinsin
sin2
1sin
2
1cossin
cos2
1cos
2
1coscos
sin2
1sin
2
1sincos
Table T.3 on pages 851-853
• Restating
• Using an Identity
• After Low Pass Filtering
58
Mixing (2)
LOLORFRFRFLO tftfAAtIF 2cos2cos
LORFLORFRFLO
LORFLORFRFLO
tffAA
tffAAtIF
2cos2
1
2cos2
1
LORFLORFRFLO tffAAtIF 2cos2
1
59
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
60
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
61
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 Publicationshttps://www.rfcafe.com/references/articles/wj-tech-notes/watkins_johnson_tech-notes.htm
Watkins-Johnson (1957-2000) and later WJ Communications (2000-2008) was acquired by Triquint (1985-2014) which merged with RF Micro Devices and is
now called Qorvo (RF Solutions for 5G and beyond)https://www.qorvo.com/
https://www.qorvo.com/products
62
Convolution• Filtering of unwanted spectral components
is performed by filtering. – Convolution in the time domain
– Multiplication in the frequency domain
63
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Graphical interpretation of convolution
dtwvtwtv
64
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Result of the convolution
65
Impulses and Transformsin the Limit
• When dealing with discrete, inherently discontinuous message data we require appropriate mathematical methods to derive and describe the modulated waveforms.
• Signal descriptions for impulses (in time and frequency), step functions, etc. are required.– Define a continuous time, parameterized function that
approaches an impulse/step function as one of the parameters approaches infinity or zero.
– What are some of these functions.
66
Delta Function Approximations• Rect
• Sinc
• Gaussian
67
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Two functions that become impulses as 0
68
Impulse Properties• Continuous sampling is equivalent to discrete
samples
• Scaling
ddd tttvtttv
dd tvdttttv
ta
ta 1
69
Impulses in Frequency
• Transform pairs
W
f
W
Aftv
W 22lim
0
AtWAtvW
2sinclim0
W
f
WtW
F
22
12sinc
fAAF
AtAF
70
Signal Smoothing• Signal approximations that provide rounding or
smoothing of rapid transitions in time ….
• Inherent in transmitted signals due to component and channel effects …
• Usually consider are low-pass filter functions … like an R-C filter previously discussed.
71
Smoothing the Edges
• A more practical frequency domain filter:The raised Cosine filter– Cosine band edge roll-off is often used
– Easy to implement in MATLAB
• A nice explanation of “filter roll-off” is provided
72
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Raised cosine pulse. (a) Waveform (b) Derivatives(c) Amplitude spectrum
Figure 2.5-7