signal and linear system analysis and linear system analysis contents 2.1 signal models ... a signal...

Chapter 2Signal and Linear System Analysis

Contents

2.1 Signal Models . . . . . . . . . . . . . . . . . . . . . 2-32.1.1 Deterministic and Random Signals . . . . . . 2-32.1.2 Periodic and Aperiodic Signals . . . . . . . . 2-32.1.3 Phasor Signals and Spectra . . . . . . . . . . 2-42.1.4 Singularity Functions . . . . . . . . . . . . . 2-7

2.2 Signal Classifications . . . . . . . . . . . . . . . . . 2-112.3 Generalized Fourier Series . . . . . . . . . . . . . . 2-142.4 Fourier Series . . . . . . . . . . . . . . . . . . . . . 2-20

2.4.1 Complex Exponential Fourier Series . . . . . 2-202.4.2 Symmetry Properties of the Fourier Coeffi-

cients . . . . . . . . . . . . . . . . . . . . . 2-232.4.3 Trigonometric Form . . . . . . . . . . . . . 2-252.4.4 Parseval’s Theorem . . . . . . . . . . . . . . 2-262.4.5 Line Spectra . . . . . . . . . . . . . . . . . 2-262.4.6 Numerical Calculation of Xn . . . . . . . . . 2-302.4.7 Other Fourier Series Properties . . . . . . . . 2-37

2.5 Fourier Transform . . . . . . . . . . . . . . . . . . . 2-38

2-1

CONTENTS

2.5.1 Amplitude and Phase Spectra . . . . . . . . 2-392.5.2 Symmetry Properties . . . . . . . . . . . . . 2-392.5.3 Energy Spectral Density . . . . . . . . . . . 2-402.5.4 Transform Theorems . . . . . . . . . . . . . 2-422.5.5 Fourier Transforms in the Limit . . . . . . . 2-512.5.6 Fourier Transforms of Periodic Signals . . . 2-532.5.7 Poisson Sum Formula . . . . . . . . . . . . 2-59

2.6 Power Spectral Density and Correlation . . . . . . . 2-602.6.1 The Time Average Autocorrelation Function 2-612.6.2 Power Signal Case . . . . . . . . . . . . . . 2-622.6.3 Properties of R.�/ . . . . . . . . . . . . . . 2-63

2.7 Linear Time Invariant (LTI) Systems . . . . . . . . . 2-742.7.1 Stability . . . . . . . . . . . . . . . . . . . . 2-762.7.2 Transfer Function . . . . . . . . . . . . . . . 2-762.7.3 Causality . . . . . . . . . . . . . . . . . . . 2-772.7.4 Properties of H.f / . . . . . . . . . . . . . . 2-782.7.5 Input/Output with Spectral Densities . . . . . 2-822.7.6 Response to Periodic Inputs . . . . . . . . . 2-822.7.7 Distortionless Transmission . . . . . . . . . 2-832.7.8 Group and Phase Delay . . . . . . . . . . . . 2-842.7.9 Nonlinear Distortion . . . . . . . . . . . . . 2-872.7.10 Ideal Filters . . . . . . . . . . . . . . . . . . 2-892.7.11 Realizable Filters . . . . . . . . . . . . . . . 2-912.7.12 Pulse Resolution, Risetime, and Bandwidth . 2-101

2.8 Sampling Theory . . . . . . . . . . . . . . . . . . . 2-1062.9 The Hilbert Transform . . . . . . . . . . . . . . . . 2-1062.10 The Discrete Fourier Transform and FFT . . . . . . . 2-106

2-2 ECE 5625 Communication Systems I

2.1. SIGNAL MODELS

2.1 Signal Models

2.1.1 Deterministic and Random Signals

� Deterministic Signals, used for this course, can be modeled ascompletely specified functions of time, e.g.,

x.t/ D A.t/ cosŒ2�f0.t/t C �.t/�

– Note that here we have also made the amplitude, fre-quency, and phase functions of time

– To be deterministic each of these functions must be com-pletely specified functions of time

� Random Signals, used extensively in Comm Systems II, takeon random values with known probability characteristics, e.g.,

x.t/ D x.t; �i/

where �i corresponds to an elementary outcome from a samplespace in probability theory

– The �i create a ensemble of sample functions x.t; �i/, de-pending upon the outcome drawn from the sample space

2.1.2 Periodic and Aperiodic Signals

� A deterministic signal is periodic if we can write

x.t C nT0/ D x.t/

for any integer n, with T0 being the signal fundamental period

ECE 5625 Communication Systems I 2-3

CONTENTS

� A signal is aperiodic otherwise, e.g.,

….t/ D

(1; jt j � 1=2

0; otherwise

(a) periodic signal, (b) aperiodic signal, (c) random signal

2.1.3 Phasor Signals and Spectra

� A complex sinusoid can be viewed as a rotating phasor

Qx.t/ D Aej.!0tC�/; �1 < t <1

� This signal has three parameters, amplitude A, frequency !0,and phase �

� The fixed phasor portion is Aej� while the rotating portion isej!0t


2.1. SIGNAL MODELS

� This signal is periodic with period T0 D 2�=!0

� It also related to the real sinusoid signal A cos.!0t C �/ viaEuler’s theorem

x.t/ D Re˚Qx.t/

D Re

˚A cos.!0t C �/C jA sin.!0t C �/

D A cos.!0t C �/

(a) obtain x.t/ from Qx.t/, (b) obtain x.t/ from Qx.t/ and Qx�.t/

� We can also turn this around using the inverse Euler formula

x.t/ D A cos.!0t C �/

D1

2Qx.t/C

1

2Qx�.t/

DAej.!0tC�/ C Ae�j.!0tC�/

2

� The frequency spectra of a real sinusoid is the line spectra plot-ted in terms of the amplitude and phase versus frequency


CONTENTS

� The relevant parameters are A and � for a particular f0 D!0=.2�/

(a) Single-sided line spectra, (b) Double-sided line spectra

� Both the single-sided and double-sided line spectra, shownabove, correspond to the real signal x.t/ D A cos.2�f0t C �/

Example 2.1: Multiple Sinusoids

� Suppose that

x.t/ D 4 cos.2�.10/t C �=3/C 24 sin.2�.100/t � �=8/

� Find the two-sided amplitude and phase line spectra of x.t/

� First recall that cos.!0t � �=2/ D sin.!0t /, so

x.t/ D 4 cos.2�.10/t C �=3/C 24 cos.2�.100/t � 5�=8/

� The complex sinusoid form is directly related to the two-sidedline spectra since each real sinusoid is composed of positiveand negative frequency complex sinusoids

x.t/ D 2hej.2�.10/tC�=3/ C e�j.2�.10/tC�=3/

iC 12

hej.2�.100/t�5�=8/ C e�j.2�.100/t�5�=8/

i2-6 ECE 5625 Communication Systems I

2.1. SIGNAL MODELS

f (Hz)

f (Hz)

Am

plitu

dePh

ase

10

12

2

100-100 -10

5π/8

-5π/8-π/3

π/3

Two-sided amplitude and phase line spectra

2.1.4 Singularity Functions

Unit Impulse (Delta) Function

� Singularity functions, such as the delta function and unit step

� The unit impulse function, ı.t/ has the operational propertiesZ t2

t1

ı.t � t0/ dt D 1; t1 < t0 < t2

ı.t � t0/ D 0; t ¤ t0

which implies that for x.t/ continuous at t D t0, the siftingproperty holds Z

1

�1

x.t/ı.t � t0/ dt D x.t0/

– Alternatively the unit impulse can be defined asZ1

�1

x.t/ı.t/ dt D x.0/


CONTENTS

� Properties:

1. ı.at/ D ı.t/=jaj

2. ı.�t / D ı.t/

3. Sifting property special cases

Z t2

t1

x.t/ı.t � t0/ dt D

8̂̂<̂:̂x.t0/; t1 < t0 < t2

0; otherwise

undefined; t0 D t1 or t0 D t2

4. Sampling property

x.t/ı.t � t0/ D x.t0/ı.t � t0/

for x.t/ continuous at t D t05. Derivative propertyZ t2

t1

x.t/ı.n/.t � t0/ dt D .�1/nx.n/.t0/

D .�1/nd nx.t/

dtn

ˇ̌̌̌tDt0

Note: Dealing with the derivative of a delta function re-quires care

� A test function for the unit impulse function helps our intuitionand also helps in problem solving

� Two functions of interest are

ı�.t/ D1

2�…

�t

2�

�D

(12�; jt j � �

0; otherwise

ı1�.t/ D �

�1

�tsin�t

�

�22-8 ECE 5625 Communication Systems I

2.1. SIGNAL MODELS

Test functions for the unit impulse ı.t/: (a) ı�.t/, (b) ı1�.t/

� In both of the above test functions letting � ! 0 results in afunction having the properties of a true delta function

Unit Step Function

� The unit step function can be defined in terms of the unit im-pulse

u.t/ �

Z t

�1

ı.�/ d� D

8̂̂<̂:̂0; t < 0

1; t > 0

undefined; t D 0

alsoı.t/ D

du.t/

dt


CONTENTS

Example 2.2: Unit Impulse 1st-Derivative

� Consider Z1

�1

x.t/ı0.t/ dt

� Using the rectangular pulse test function, ı�.t/, we note that

ı�.t/ D1

2�…

�t

2�

�alsoD

1

2�

�u.t C �/ � u.t � �/

�and

dı�.t/

dtD

1

2�

�ı.t C �/ � ı.t � �/

�� Placing the above in the integrand with x.t/ we obtain, with

the aid of the sifting property, thatZ1

�1

x.t/ı0.t/ dt D lim�!0

1

2�

�x.t C �/ � x.t � �/

�D lim

�!0

��x.t � �/ � x.t C �/

�2�

D �x0.0/


2.2. SIGNAL CLASSIFICATIONS

2.2 Signal Classifications

� From circuits and systems we know that a real voltage or cur-rent waveform, e.t/ or i.t/ respectively, measured with respec-tive a real resistance R, the instantaneous power is

P.t/ D e.t/i.t/ D i2.t/R W

� On a per-ohm basis, we obtain

p.t/ D P.t/=R D i2.t/ W/ohm

� The average energy and power can be obtain by integratingover the interval jt j � T with T !1

E D limT!1

Z T

�T

i2.t/ dt Joules/ohm

P D limT!1

1

2T

Z T

�T

i2.t/ dt W/ohm

� In system engineering we take the above energy and powerdefinitions, and extend them to an arbitrary signal x.t/, pos-sibly complex, and define the normalized energy (e.g. 1 ohmsystem) as

E�D lim

T!1

Z T

�T

jx.t/j2 dt D

Z1

�1

jx.t/j2 dt

P�D lim

T!1

1

2T

Z T

�T

jx.t/j2 dt


CONTENTS

� Signal Classes:

1. x.t/ is an energy signal if and only if 0 < E <1 so thatP D 0

2. x.t/ is a power signal if and only if 0 < P < 1 whichimplies that E !1

Example 2.3: Real Exponential

� Consider x.t/ D Ae�˛tu.t/ where ˛ is real

� For ˛ > 0 the energy is given by

E D

Z1

0

�Ae�˛t

�2dt D

A2e�2˛t

�2˛

ˇ̌̌̌1

0

DA2

2˛

� For ˛ D 0 we just have x.t/ D Au.t/ and E !1

� For ˛ < 0 we also have E !1

� In summary, we conclude that x.t/ is an energy signal for ˛ >0

� For ˛ > 0 the power is given by

P D limT!1

1

2T

A2

2˛

�1 � e�˛T

�D 0

� For ˛ D 0 we have

P D limT!1

1

2T� A2T D

A2

2


2.2. SIGNAL CLASSIFICATIONS

� For ˛ < 0 we have P !1

� In summary, we conclude that x.t/ is a power signal for ˛ D 0

Example 2.4: Real Sinusoid

� Consider x.t/ D A cos.!0t C �/; �1 < t <1

� The signal energy is infinite since upon squaring, and integrat-ing over one cycle, T0 D 2�=!0, we obtain

E D limN!1

Z NT0=2

�NT0=2

A2 cos2.!0t C �/ dt

D limN!1

N

Z T0=2

�T0=2

A2 cos2.!0t C �/ dt

D limN!1

NA2

2

Z T0=2

�T0=2

�1C cos.2!0t C 2�/ dt

D limN!1

NA2

2� T0!1

� The signal average power is finite since the above integral isnormalized by 1=.NT0/, i.e.,

P D limN!1

1

NT0�N

A2

2� T0 D

A2

2


CONTENTS

2.3 Generalized Fourier Series

The goal of generalized Fourier series is to obtain a representation ofa signal in terms of points in a signal space or abstract vector space.The coordinate vectors in this case are orthonomal functions. Thecomplex exponential Fourier series is a special case.

� Let EA be a vector in a three dimensional vector space

� Let Ea1; Ea2, and Ea3 be linearly independent vectors in the samethree dimensional space, then

c1Ea1 C c2Ea2 C c3Ea3 D 0 .zero vector/

only if the constants c1 D c2 D c3 D 0

� The vectors Ea1; Ea2, and Ea3 also span the three dimensionalspace, that is for any vector EA there exists a set of constantsc1; c2, and c3 such that

EA D c1Ea1 C c2Ea2 C c3Ea3

� The set fEa1; Ea2; Ea3g forms a basis for the three dimensionalspace

� Now let fEa1; Ea2; Ea3g form an orthogonal basis, which impliesthat

Eai � Eaj D .Eai ; Eaj / D hEai ; Eaj i D 0; i ¤ j

which says the basis vectors are mutually orthogonal

� From analytic geometry (and linear algebra), we know that wecan find a representation for EA as

EA D.Ea1 � EA/

jEa1j2C.Ea2 � EA/

jEa2j2C.Ea3 � EA/

jEa3j2


2.3. GENERALIZED FOURIER SERIES

which implies that

EA D

3XiD1

ci Eai

where

ci DEai � EA

jEai j2; i D 1; 2; 3

is the component of EA in the Eai direction

� We now extend the above concepts to a set of orthogonal func-tions f�1.t/; �2.t/; : : : ; �N .t/g defined on to � t � t0 C T ,where the dot product (inner product) associated with the �n’sis

��m.t/; �n.t/

�D

Z t0CT

t0

�m.t/��

n.t/ dt

D cnımn D

(cn; n D m

0; n ¤ m

� The �n’s are thus orthogonal on the interval Œt0; t0 C T �

� Moving forward, let x.t/ be an arbitrary function on Œt0; t0CT �,and consider approximating x.t/ with a linear combination of�n’s, i.e.,

x.t/ ' xa.t/ D

NXnD1

Xn�n.t/; t0 � t � t0 C T;

where a denotes approximation


CONTENTS

� A measure of the approximation error is the integral squarederror (ISE) defined as

�N D

ZT

ˇ̌x.t/ � xa.t/

ˇ̌2dt;

whereRT

denotes integration over any T long interval

� To find the Xn’s giving the minimum �N we expand the aboveintegral into three parts (see homework problems)

�N D

ZT

jx.t/j2 dt �

NXnD1

1

cn

ˇ̌̌̌ZT

x.t/��n.t/ dt

ˇ̌̌̌2C

NXnD1

cn

ˇ̌̌̌Xn �

1

cn

ZT

x.t/��n.t/ dt

ˇ̌̌̌2– Note that the first two terms are independent of the Xn’s

and the last term is nonnegative (missing steps are in texthomework problem 2.14)

� We conclude that �N is minimized for each n if each elementof the last term is made zero by setting

Xn D1

cn

ZT

x.t/��n.t/ dt Fourier Coefficient

� This also results in

��N�

min D

ZT

jx.t/j2 dt �

NXnD1

cnjXnj2



� Definition: The set of of �n’s is complete if

limN!1

.�N /min D 0

forRTjx.t/j2 dt <1

– Even if though the ISE is zero when using a completeset of orthonormal functions, there may be isolated pointswhere x.t/ � xa.t/ ¤ 0

� Summary

x.t/ D l.i.m.1XnD1

Xn�n.t/

Xn D1

cn

ZT

x.t/��n.t/ dt

– The notation l.i.m. stands for limit in the mean, which isa mathematical term referring to the fact that ISE is theconvergence criteria

� Parseval’s theorem: A consequence of completeness isZT

jx.t/j2 dt D

1XnD1

cnjXnj2


CONTENTS

Example 2.5: A Three Term Expansion

� Approximate the signal x.t/ D cos 2�t on the interval Œ0; 1�using the following basis functions

0.2 0.4 0.6 0.8 1

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

0.2 0.4 0.6 0.8 1

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1x(t) φ

1(t)

t t

0.2 0.4 0.6 0.8 1

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

0.2 0.4 0.6 0.8 1

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

φ2(t) φ

3(t)

t t

Signal x.t/ and basis functions �i.t/; i D 1; 2; 3

� To begin with it should be clear that �1.t/; �2.t/, and �3.t/are mutually orthogonal since the integrand associated with theinner product, �i.t/ � ��j .t/ D 0, for i ¤ j; i; j D 1; 2; 3



� Before finding the Xn’s we need to find the cn’s

c1 D

ZT

j�1.t/j2 dt

Z 1=4

0

j1j2 dt D 1=4

c2 D

ZT

j�2.t/j2 dt D 1=2

c3 D

ZT

j�3.t/j2 dt D 1=4

� Now we can compute the Xn’s:

X1 D 4

ZT

x.t/��1 .t/ dt

D 4

Z 1=4

0

cos.2�t/ dt D2

�sin.2�t/

ˇ̌̌1=40D2

�

X2 D 2

Z 3=4

1=4

cos.2�t/ dt D1

�sin.2�t/

ˇ̌̌3=41=4D�2

�

X3 D 4

Z 1

3=4

cos.2�t/ dt D2

�sin.2�t/

ˇ̌̌13=4D2

�

t

x(t)

xa(t)

0.2 0.4 0.6 0.8 1

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

-2/π

2/π

Functional approximation


CONTENTS

� The integral-squared error, �N , can be computed as follows:

�N D

ZT

ˇ̌̌̌x.t/ �

3XnD1

Xn�n.t/

ˇ̌̌̌2dt

D

ZT

jx.t/j2 dt �

3XnD1

cnjXnj2

D1

2�1

4

ˇ̌̌̌2

�

ˇ̌̌̌2�1

2

ˇ̌̌̌2

�

ˇ̌̌̌2�1

4

ˇ̌̌̌2

�

ˇ̌̌̌2D1

2�

ˇ̌̌̌2

�

ˇ̌̌̌2D 0:0947

2.4 Fourier Series

When we choose a particular set of basis functions we arrive at themore familiar Fourier series.

2.4.1 Complex Exponential Fourier Series

� A set of �n’s that is complete is

�n.t/ D ejn!0t ; n D 0;˙1;˙2; : : :

over the interval .t0; t0 C T0/ where !0 D 2�=T0 is the periodof the expansion interval


2.4. FOURIER SERIES

proof of orthogonality��m.t/; �n.t/

�D

Z t0CT0

t0

ejm2�tT0 e

�jn2�tT0 dt D

Z t0CT0

t0

ej 2�T0

.m�n/tdt

D

8̂̂<̂:̂R t0CT0t0

dt; m D nR t0CT0t0

�cosŒ2�.m � n/t=T0�

Cj sinŒ2�.m � n/t=T0��dt; m ¤ n

D

(T0; m D n

0; m ¤ n

We also conclude that cn D T0

� Complex exponential Fourier series summary:

x.t/ D

1XnD�1

Xnejn!0t ; t0 � t � t0 C T0

where Xn D1

T0

ZT0

x.t/e�jn!0t

� The Fourier series expansion is unique

Example 2.6: x.t/ D cos2!0t

� If we expand x.t/ into complex exponentials we can immedi-ately determine the Fourier coefficients

x.t/ D1

2C1

2cos 2!0t

D1

2C1

4ej 2!0t C

1

4e�j 2!0t


CONTENTS

� The above implies that

Xn D

8̂̂<̂:̂12; n D 014; n D ˙2

0; otherwise

Time Average Operator

� The time average of signal v.t/ is defined as

hv.t/i�D lim

T!1

1

2T

Z T

�T

v.t/ dt

� Note that

hav1.t/C bv2.t/i D ahv1.t/i C bhv2.t/i;

where a and b are arbitrary constants

� If v.t/ is periodic, with period T0, then

hv.t/i D1

T0

ZT0

v.t/ dt

� The Fourier coefficients can be viewed in terms of the timeaverage operator

� Let v.t/ D x.t/e�jn!0t using e�j� D cos � � j sin � , we findthat

Xn D hv.t/i D hx.t/e�jn!0ti

D hx.t/ cosn!0ti � j hx.t/ sinn!0ti


2.4. FOURIER SERIES

2.4.2 Symmetry Properties of the Fourier Coef-ficients

� For x.t/ real, the following coefficient symmetry propertieshold:

1. X�n D X�n2. jXnj D jX�nj

3. †Xn D �†X�n

proof

X�n D

�1

T0

ZT0

x.t/e�jn!0t dt

��D

1

T0

ZT0

x.t/e�j.�n/!0t dt D X�n

since x�.t/ D x.t/

� Waveform symmetry conditions produce special results too

1. If x.�t / D x.t/ (even function), then

Xn D Re˚Xn; i.e., Im

˚XnD 0

2. If x.�t / D �x.t/ (odd function), then

Xn D Im˚Xn; i.e., Re

˚XnD 0

3. If x.t ˙ T0=2/ D �x.t/ (odd half-wave symmetry), then

Xn D 0 for n even


CONTENTS

Example 2.7: Odd Half-wave Symmetry Proof

� Consider

Xn D1

T0

Z t0DT0=2

t0

x.t/e�jn!0t dtC1

T0

Z t0CT0

t0CT0=2

x.t 0/e�jn!0t0

dt 0

� In the second integral we change variables by letting t D t 0 �

T0=2

Xn D1

T0

Z t0CT0=2

t0

x.t/e�jn!0t dt

C1

T0

Z tCT0=2

t0

x.t � T0=2/„ ƒ‚ …�x.t/

e�jn!0.tCT0=2/ dt

D

�1 � e�jn!0T0=2

� 1T0

Z t0CT0=2

t0

x.t/e�jn!0t dt

but n!0.T0=2/ D n.2�=T0/.T0=2/ D n� , thus

1 � e�jn� D

(2; n odd

0; n even

� We thus see that the even indexed Fourier coefficients are in-deed zero under odd half-wave symmetry


2.4. FOURIER SERIES

2.4.3 Trigonometric Form

� The complex exponential Fourier series can be arranged as fol-lows

x.t/ D

1XnD�1

Xnejn!0t

D X0 C

1XnD1

�Xne

jn!0t CX�ne�jn!0t

�� For real x.t/, we may know that jX�nj D jXnj and †Xn D�†X�n, so

x.t/ D X0 C

1XnD1

�jXnje

j Œn!0tC†Xn� C jXnje�j Œn!0tC†Xn�

�D X0 C 2

1XnD1

jXnj cos�n!0t C†Xn

�since cos.x/ D .ejx C e�jx/=2

� From the trig identity cos.uC v/ D cosu cos v� sinu sin v, itfollows that

x.t/ D X0 C

1XnD1

An cos.n!0t /C1XnD1

Bn sin.n!0t /

where

An D 2hx.t/ cos.n!0t /iBn D 2hx.t/ sin.n!0t /i


CONTENTS

2.4.4 Parseval’s Theorem

� Fourier series analysis are generally used for periodic signals,i.e., x.t/ D x.t C nT0/ for any integer n

� With this in mind, Parseval’s theorem becomes

P D1

T0

ZT0

jx.t/j2 dt D

1XnD�1

jXnj2

D X20 C 2

1XnD1

jXnj2 .W/

Note: A 1 ohm system is assumed

2.4.5 Line Spectra

� Line spectra was briefly reviewed earlier for simple signals

� For any periodic signal having Fourier series representation wecan obtain both single-sided and double-sided line spectra

� The double-sided magnitude and phase line spectra is mosteasily obtained form the complex exponential Fourier series,while the single-sided magnitude and phase line spectra can beobtained from the trigonometric form

Double-sidedmag. and phase

()

1XnD�1

Xnej 2�.nf0/t

Single-sidedmag. and phase

() X0 C 2

1XnD1

jXnj cosŒ2�.nf0/t C†Xn�


2.4. FOURIER SERIES

– For the double-sided simply plot as lines jXnj and †Xnversus nf0 for n D 0;˙1;˙2; : : :

– For the single-sided plot jX0j and †X0 as a special casefor n D 0 at nf0 D 0 and then plot 2jXnj and †X0 versusnf0 for n D 1; 2; : : :

Example 2.8: Cosine Squared

� Consider

x.t/ D A cos2.2�f0t C �/ DA

2CA

2cos

�2�.2f0/t C 2�1

�DA

2CA

4ej 2�1ej 2�.2f0/t C

A

4e�j 2�1e�j 2�.2f0/t

Double-Sided

Single-Sided

-2f0

-2f0

2f0

2f0

2f0

2f0

f

f f

f


CONTENTS

Example 2.9: Pulse Train

t

A

x(t)

0 τ T0T

0 + τ-T

0-2T

0

. . .. . .

Periodic pulse train

� The pulse train signal is mathematically described by

x.t/ D

1XnD�1

A…

�t � nT0 � �=2

�

�

� The Fourier coefficients are

Xn D1

T0

Z �

0

Ae�j 2�.nf0/t dt DA

T0�e�j 2�.nf0/t

�j 2�.nf0/

ˇ̌̌�0

DA

T0�1 � e�j 2�.nf0/�

j 2�.nf0/

DA�

T0�ej�.nf0/� � e�j�.nf0/�

.2j /�.nf0/�� e�j�.nf0/�

DA�

T0�

sinŒ�.nf0/��Œ�.nf0/��

� e�j�.nf0/�

� To simplify further we define

sinc.x/ �D

sin.�x/�x


2.4. FOURIER SERIES

� Finally,

Xn DA�

T0sinc.nf0�/e�j�.nf0/� ; n D 0;˙1;˙2; : : :

� To plot the line spectra we need to find jXnj and †Xn

jXnj DA�

T0jsincŒ.nfo/��j

†Xn D

8̂̂<̂:̂��.nf0/�; sinc.nfo�/ > 0

��.nf0/� C �; nf0 > 0 and sinc.nf0�/ < 0

��.nf0/� � �; nf0 < 0 and sinc.nf0�/ < 0

� Pulse train double-sided line spectra for � D 0:125 using Python,specifically use ssd.line_spectra()

� As a specific example enter the following at the IPython com-mand prompt

n = arange(0,25+1) # Get 0 through 25 harmonicstau = 0.125; f0 = 1; A = 1;Xn = A*tau*f0*sinc(n*f0*tau)*exp(-1j*pi*n*f0*tau)figure(figsize=(6,2))f = n # Assume a fundamental frequency of 1 Hz so f = nssd.line_spectra(f,Xn,mode=’mag’,fsize=(6,2))xlim([-25,25]);figure(figsize=(6,2))ssd.line_spectra(f,Xn,mode=’phase’,fsize=(6,2))xlim([-25,25]);


CONTENTS

0 0.125fAt = 0 1, 0.125f t ==

1 8t =

Phase slope0.125

ff

p tp

-=- ´=

2.4.6 Numerical Calculation of Xn

� Here we consider a purely numerical calculation of the Xk co-efficients from a single period waveform description of x.t/

� In particular, we will use the numpy fast Fourier transform(FFT) function to carry out the numerical integration

� By definition

Xk D1

T0

ZT0

x.t/e�j 2�kf0t dt; k D 0;˙1;˙2; : : :


2.4. FOURIER SERIES

� A simple rectangular integration approximation to the aboveintegral is

Xk '1

T0

N�1XnD0

x.nT /e�jk2�.nf0/T0=N �T0

N; k D 0;˙1;˙2; : : :

where N is the number of points used to partition the timeinterval Œ0; T0� and T D T0=N is the time step

� Using the fact that 2�f0T0 D 2� , we can write that

Xk '1

N

N�1XnD0

x.nT /e�j 2�kn

N ; k D 0;˙1;˙2; : : :

� Note that the above must be evaluated for each Fourier coeffi-cient of interest

� Also note that the accuracy of the Xk values depends on thevalue of N

– For k small and x.t/ smooth in the sense that the harmon-ics rolloff quickly,N on the order of 100 may be adequate

– For k moderate, say 5–50, N will have to become in-creasingly larger to maintain precision in the numericalintegral

Calculation Using the FFT

� The FFT is a powerful digital signal processing (DSP) func-tion, which is a computationally efficient version of the dis-crete Fourier transform (DFT)


CONTENTS

� For the purposes of the problem at hand, suffice it to say thatthe FFT is just an efficient algorithm for computing

XŒk� D

N�1XnD0

xŒn�e�j 2�kn=N ; k D 0; 1; 2; : : : ; N � 1

� If we let xŒn� D x.nT /, then it should be clear that

Xk '1

NXŒk�; k D 0; 1; : : : ;

N

2

� To obtain Xk for k < 0 note that

X�k '1

NXŒ�k� D

1

N

N�1XnD0

x.nT /e�j 2�.�k/n

N

D1

N

N�1XnD0

x.nT /e�j 2�.N�k/n

N D XŒN � k�

since e�j 2�Nn=N D e�j 2�n D 1

� In summary

Xk '

(XŒk�=N; 0 � k � N=2

XŒN � k�=N; �N=2 � k < 0

� To use the Python function fft.fft() to obtain the Xk wesimply let

X D fft.fft(x)

where xD fx.t/ W t D 0; T0=N; 2T0=N; : : : ; T0.N � 1/=N g

� Unlike in MATLAB XŒ0� is really found in X[0]


2.4. FOURIER SERIES

Example 2.10: Finite Rise/Fall-Time PulseTrain

t

x(t)1

1/2

0 tr T

0τ

τ

τ + tr

Pulse width = τRise and fall time = t

r

Pulse train with finite rise and fall time edges

� Shown above is one period of a finite rise and fall time pulsetrain

� We will numerically compute the Fourier series coefficients ofthis signal using the FFT

� The Python function trap_pulse was written to generate oneperiod of the signal using N samples

def trap_pulse(N,tau,tr):"""xp = trap_pulse(N,tau,tr)

Mark Wickert, January 2015"""n = arange(0,N)t = n/Nxp = zeros(len(t))# Assume tr and tf are equalT1 = tau + tr# Create one period of the trapezoidal pulse waveformfor k in n:

if t[k] <= tr:xp[k] = t[k]/tr

elif (t[k] > tr and t[k] <= tau):


CONTENTS

xp[k] = 1elif (t[k] > tau and t[k] < T1):

xp[k] = -t[k]/tr + 1 + tau/tr;else:

xp[k] = 0return xp, t

� We now plot the double-sided line spectra for � D 1=8 and twovalues of rise-time tr


2.4. FOURIER SERIES

1/20

1/8

1/τ = 8

Sidelobes smaller than ideal pulse train which has zero rise time

Signal x.t/ and line spectrum for � D 1=8 and tr D 1=20

� The spectral roll-off rate is faster with the trapezoid

� Clock edges are needed in digital electronics, but by slowingthe edge speed down the clock harmonic generation can bereduced

� The second plot is a more extreme example


CONTENTS

1/10

1/8

1/τ = 8Sidelobes smaller than with t

r = 1/20

case ~10dB more

Signal x.t/ and line spectrum for � D 1=8 and tr D 1=10


2.4. FOURIER SERIES

2.4.7 Other Fourier Series Properties

� Given x.t/ has Fourier series (FS) coefficients Xn, if

y.t/ D AC Bx.t/

it follows that

Yn D

(AC BX0; n D 0

BXn; n ¤ 0

proof:

Yn D hy.t/e�j 2�.nf0/ti

D Ahe�j 2�.nf0/ti C Bhx.t/e�j 2�.nf0/ti

D A

�1; n D 0

0; n ¤ 0

�C BXn

QED

� Likewise ify.t/ D x.t � t0/

it follows thatYn D Xne

�j 2�.nf0/t0

proof:Yn D hx.t � t0/e

�j 2�.nf0/ti

Let � D t � t0 which implies also that t D �C t0, so

Yn D hx.�/e�j 2�.nf0/.�Ct0/i

D hx.�/e�j 2�.nf0/�ie�j 2�.nf0/t0

D Xne�j 2�.nf0/t0

QED


CONTENTS

2.5 Fourier Transform

� The Fourier series provides a frequency domain representationof a periodic signal via the Fourier coefficients and line spec-trum

� The next step is to consider the frequency domain representa-tion of aperiodic signals using the Fourier transform

� Ultimately we will be able to include periodic signals withinthe framework of the Fourier transform, using the concept oftransform in the limit

� The text establishes the Fourier transform by considering alimiting case of the expression for the Fourier series coefficientXn as T0!1

� The Fourier transform (FT) and inverse Fourier transfrom (IFT)is defined as

X.f / D

Z1

�1

x.t/e�j 2�f t dt (FT)

x.t/ D

Z1

�1

X.f /ej 2�f t df (IFT)

� Sufficient conditions for the existence of the Fourier transformare

1.R1

�1jx.t/j dt <1

2. Discontinuities in x.t/ be finite

3. An alternate sufficient condition is thatR1

�1jx.t/j2 dt <

1, which implies that x.t/ is an energy signal


2.5. FOURIER TRANSFORM

2.5.1 Amplitude and Phase Spectra

� FT properties are very similar to those obtained for the Fouriercoefficients of periodic signals

� The FT, X.f / D Ffx.t/g, is a complex function of f

X.f / D jX.f /jej�.f / D jX.f /jej†X.f /

D RefX.f /g C j ImfX.f /g

� The magnitude jX.f /j is referred to as the amplitude spectrum

� The the angle †X.f / is referred to as the phase spectrum

� Note that

RefX.f /g DZ1

�1

x.t/ cos 2�f t dt

ImfX.f /g DZ1

�1

x.t/ sin 2�f t dt

2.5.2 Symmetry Properties

� If x.t/ is real it follows that

X.�f / D

Z1

�1

x.t/e�j 2�.�f /t dt

D

�Z1

�1

x.t/e�j 2�f t dt

��D X�.f /

thus

jX.�f /j D jX.f /j (even in frequency)†X.�f / D �†X.f / (odd in frequency)


CONTENTS

� Additionally,

1. For x.�t / D x.t/ (even function), ImfX.f /g D 0

2. For x.�t / D �x.t/ (odd function), RefX.f /g D 0

2.5.3 Energy Spectral Density

� From the definition of signal energy,

E D

Z1

�1

jx.t/j2 dt

D

Z1

�1

x�.t/

�Z1

�1

X.f /ej 2�f t df

�dt

D

Z1

�1

X.f /

�Z1

�1

x�.t/ej 2�f t dt

�df

butZ1

�1

x�.t/ej 2�f t dt D

�Z1

�1


��D X�.f /

� Finally,

E D

Z1

�1

jx.t/j2 dt D

Z1

�1

jX.f /j2 df

which is known as Rayleigh’s Energy Theorem

� Are the units consistent?

– Suppose x.t/ has units of volts

– jX.f /j2 has units of (volts-sec)2



– In a 1 ohm system jX.f /j2 has units of Watts-sec/Hz =Joules/Hz

� The energy spectral density is defined as

G.f /�D jX.f /j2 Joules/Hz

� It then follows that

E D

Z1

�1

G.f / df

Example 2.11: Rectangular Pulse

� Consider

x.t/ D A…

�t � t0

�

�� FT is

X.f / D A

Z t0C�=2

t0��=2

e�j 2�f t dt

D A �e�j 2�f t

�j 2�f

ˇ̌̌̌ˇt0C�=2

t0��=2

D A� �

�ej�f � � e�j�f �

.j 2/�f �

�� e�j 2�f t0

D A�sinc.f �/e�j 2�f t0

A…

�t � t0

�

�F ! A�sinc.f �/e�j 2�f t0

� Plot jX.f /j, †X.f /, and G.f /


CONTENTS

? 3 ? 2 ? 1 1 2 3

0.2

0.4

0.6

0.8

1

? 3 ? 2 ? 1 1 2 3

? 3

? 2

? 1

1

2

3

? 3 ? 2 ? 1 1 2 3

0.2

0.4

0.6

0.8

1

f

f

f

0-1/τ 1/τ 2/τ-2/τ

0-1/τ 1/τ 2/τ-2/τ

-1/τ 1/τ 2/τ-2/τ

|X(f)|

G(f) = |X(f)|2

X(f)AmplitudeSpectrum

EnergySpectralDensity

PhaseSpectrum

Aτ

(Aτ)2

π

π/2

−π/2

−πt0 = τ/2 slope = -πfτ/2

Rectangular pulse spectra

2.5.4 Transform Theorems

� Be familiar with the FT theorems found in the table of Ap-pendix G.6 of the text

Superposition Theorem

a1x1.t/C a2x2.t/F ! a1X1.f /C a2X2.f /

proof:



Time Delay Theorem

x.t � t0/F ! X.f /e�j 2�f t0

proof:

Frequency Translation Theorem

� In communications systems the frequency translation and mod-ulation theorems are particularly important

x.t/ej 2�f0tF ! X.f � f0/

proof: Note thatZ1

�1

x.t/ej 2�f0te�j 2�f t dt D

Z1

�1

x.t/e�j 2�.f �f0/t dt

soF˚x.t/ej 2�f0t

D X.f � f0/

QED

Modulation Theorem

� The modulation theorem is an extension of the frequency trans-lation theorm

x.t/ cos.2�f0t /F !

1

2X.f � f0/C

1

2X.f C f0/


CONTENTS

proof: Begin by expanding

cos.2�f0t / D1

2ej 2�f0t C

1

2e�j 2�f0t

Then apply the frequency translation theorem to each term sep-arately

x(t) y(t)

cos(2πf0t)

X(f)Y(f)

f f

A/2

A

f0

-f0

00

signalmultiplier

A simple modulator

Duality Theorem

� Note that

FfX.t/g DZ1

�1

X.t/e�j 2�f t dt D

Z1

�1

X.t/ej 2�.�f /t dt

which implies that

X.t/F ! x.�f /



Example 2.12: Rectangular Spectrum

f-W W0

1X(f)

� Using duality on the above we have

X.t/ D …

�t

2W

�F ! 2W sinc.2Wf / D x.�f /

� Since sinc( ) is an even function (sinc.x/ D sinc.�x/), it fol-lows that

2W sinc.2W t/F ! …

�f

2W

�

Differentiation Theorem

� The general result isd nx.t/

dtnF ! .j 2�f /n X.f /

proof: For n D 1we start with the integration by parts formula,Rudv D uv

ˇ̌̌�Rv du, and apply it to

F�dx

dt

�D

Z1

�1

dx

dte�j 2�f t dt

D x.t/e�j 2�f tˇ̌̌1

�1„ ƒ‚ …0

Cj 2�f

Z1

�1

x.t/e�j 2�f t dt„ ƒ‚ …X.f /


CONTENTS

alternate — From Leibnitz’s rule for differentiation of inte-grals,

d

dt

Z1

�1

F.f; t/ df D

Z1

�1

@F.f; t/

@fdf

sodx.t/

dtDd

dt

Z1

�1

X.f /ej 2�f t df

D

Z1

�1

X.f /@ej 2�f t

@tdf

D

Z1

�1

j 2�fX.f /ej 2�f t df

) dx=dtF ! j 2�fX.f / QED

Example 2.13: FT of Triangle Pulse

τ−τ 0t

1

� Note that

ττ−τ −τ

t t

1/τ

1/τ

-2/τ-1/τ



� Using the differentiation theorem for n D 2 we have that

F�ƒ

�t

�

��D

1

.j 2�f /2Fn1�ı.t C �/ �

2

�ı.t/C

1

�ı.t � �/

oD

1�ej 2�f � � 2

�C

1�e�j 2�f �

.j 2�f /2

D2 cos.2�f �/ � 2��.2�f /2

D �4 sin2.�f �/4.�f �/2

D �sinc2.f �/

ƒ

�t

�

�F ! �sinc2.f �/

Convolution and Convolution Theorem

� Before discussing the convolution theorem we need to reviewconvolution

� The convolution of two signals x1.t/ and x2.t/ is defined as

x.t/ D x1.t/ � x2.t/ D

Z1

�1

x1.�/x2.t � �/ d�

D x2.t/ � x1.t/ D

Z1

�1

x2.�/x1.t � �/ d�

� A special convolution case is ı.t � t0/

ı.t � t0/ � x.t/ D

Z1

�1

ı.� � t0/x.t � �/ d�

D x.t � �/ˇ̌�Dt0D x.t � t0/


CONTENTS

Example 2.14: Rectangular Pulse Convolution

� Let x1.t/ D x2.t/ D ….t=�/

� To evaluate the convolution integral we need to consider theintegrand by sketching of x1.�/ and x2.t ��/ on the � axis fordifferent values of t

� For this example four cases are needed for t to cover the entiretime axis t 2 .�1;1/

� Case 1: When t < � we have no overlap so the integrand iszero and x.t/ is zero

λt t + τ/2t - τ/2 0 τ/2−τ/2

x2(t - λ) x

1(λ)

No overlap for t + τ/2 < -τ/2 or t < τ

� Case 2: When �� < t < 0 we have overlap and

x.t/ D

Z1

�1

x1.�/x2.t � �/ d�

D

Z tC�=2

��=2

d� D �ˇ̌̌tC�=2��=2

D t C �=2C �=2 D � C t

Overlap begins when t + τ/2 = -τ/2 or t = -τ

λ

t + τ/20 τ/2−τ/2

x2(t - λ) x

1(λ)



� Case 3: For 0 < t < � the leading edge of x2.t � �/ is to theright of x1.�/, but the trailing edge of the pulse is still over-lapped

x.t/ D

Z �=2

t��=2

d� D �=2 � t C �=2 D � � t

λt + τ/2

t - τ/20 τ/2−τ/2

x2(t - λ)x

1(λ)

Overlap lasts until t = τ

� Case 4: For t > � we have no overlap, and like case 1, theresult is

x.t/ D 0

λt + τ/2t - τ/20 τ/2−τ/2

x2(t - λ)x

1(λ)

No overlap for t > τ

� Collecting the results

x.t/ D

8̂̂̂̂<̂ˆ̂̂:0; t < ��

� C t; �� t < 0

� � t; 0 � t < �

0; t � �

D

(� � jt j; jt j � �

0; otherwise


CONTENTS

� Final summary,

…

�t

�

��…

�t

�

�D �ƒ

�t

�

�

� Convolution Theorem: We now consider x1.t/�x2.t/ in termsof the FTZ

1

�1

x1.�/x2.t � �/ d�

D

Z1

�1

x1.�/

�Z1

�1

X2.f /ej 2�f .t��/ df

�d�

D

Z1

�1

X2.f /

�Z1

�1

x1.�/e�j 2�f � d�

�ej 2�f t df

D

Z1

�1

X1.f /X2.f /ej 2�f t df

which implies that

x1.t/ � x2.t/F ! X1.f /X2.f /

Example 2.15: Revisit ….t=�/ �….t=�/

� Knowing that….t=�/�….t=�/ D �ƒ.t=�/ in the time domain,we can follow-up in the frequency domain by writing

F˚….t=�/

� F˚….t=�/

D��sinc.f �/

�2� We have also established the transform pair

�ƒ

�t

�

�F ! �2sinc2.f �/ D �

��sinc2.f �/



or

ƒ

�t

�

�F ! �sinc2.f �/

Multiplication Theorem

� Having already established the convolution theorem, it followsfrom the duality theorem or direct evaluation, that

x1.t/ � x2.t/F ! X1.f / �X2.f /

2.5.5 Fourier Transforms in the Limit

� Thus far we have considered two classes of signals

1. Periodic power signals which are described by line spec-tra

2. Non-periodic (aperiodic) energy signals which are describedby continuous spectra via the FT

� We would like to have a unifying approach to spectral analysis

� To do so we must allow impulses in the frequency domain byusing limiting operations on conventional FT pairs, known asFourier transforms-in-the-limit

– Note: The corresponding time functions have infinite en-ergy, which implies that the concept of energy spectraldensity will not apply for these signals (we will introducethe concept of power spectral density for these signals)


CONTENTS

Example 2.16: A Constant Signal

� Let x.t/ D A for �1 < t <1

� We can writex.t/ D lim

T!1A….t=T /

� Note thatF˚A….t=T /

D AT sinc.f T /

� Using the transform-in-the-limit approach we write

Ffx.t/g D limT!1

AT sinc.f T /

? 3 ? 2 ? 1 1 2 3? 0.2

0.2

0.4

0.6

0.8

1

? 3 ? 2 ? 1 1 2 3? 0.2

0.2

0.4

0.6

0.8

1AT1

AT2

T2 >> T

1

f f

Increasing T in AT sinc.f T /

� Note that since x.t/ has no time variation it seems reasonablethat the spectral content ought to be confined to f D 0

� Also note that it can be shown thatZ1

�1

AT sinc.f T / df D A; 8 T

� Thus we have established that

AF ! Aı.f /



� As a further check

F�1˚Aı.f /

D

Z1

�1

Aı.f /ej 2�ff t df D Aej 2�f tˇ̌̌fD0D A

� As a result of the above example, we can obtain several moreFT-in-the-limit pairs

Aej 2�f0tF ! Aı.f � f0/

A cos.2�f0t C �/F !

A

2

�ej�ı.f � f0/C e

�j�ı.f C f0/�

Aı.t � t0/F ! Ae�j 2�f t0

� Reciprocal Spreading Property: Compare

Aı.t/F ! A and A

F ! Aı.f /

A constant signal of infinite duration has zero spectral width,while an impulse in time has zero duration and infinite spectralwidth

2.5.6 Fourier Transforms of Periodic Signals

� For an arbitrary periodic signal with Fourier series

x.t/ D

1XnD�1

Xnej 2�nf0t


CONTENTS

we can write

X.f / D F"1X

nD�1

Xnej 2�nf0t

#

D

1XnD�1

XnFnej 2�nf0t

oD

1XnD�1

Xnı.f � nf0/

using superposition and FfAej 2�f0tg D Aı.f � f0/

� What is the difference between line spectra and continuousspectra? none!

� Mathematically,

LineSpectra

Convert to time domain

Convert to time domain

Sum phasors

Integrate impulses to get phasors via the inverse FT

ContinuousSpectra

� The Fourier series coefficients need to be known before the FTspectra can be obtained

� A technique that obtains the FT directly will be discussed shortly(page 2–58))



Example 2.17: Ideal Sampling Waveform

� When we discuss sampling theory it will be useful to have theFT of the periodic impulse train signal

ys.t/ D

1XmD�1

ı.t �mTs/

where Ts is the sample spacing or period

� Since this signal is periodic, it must have a Fourier series rep-resentation too

� In particular

Yn D1

Ts

ZTs

ı.t/e�j 2�.nfs/t dt D1

TsD fs; any n

where fs is the sampling rate in Hz

� The FT of ys.t/ is given by

Ys.f / D fs

1XnD�1

F˚ej 2�nf0/t

D fs

1XnD�1

ı.f � nfs/

� Summary,

1XmD�1

ı.t �mTs/F ! fs

1XnD�1

ı.f � nfs/


CONTENTS

. . . . . .

t

f

ys(t)

Ys(f)

0

0

Ts

fs 4f

s

4Ts

-Ts

-fs

. . . . . .fs

1

An impulse train in time is an impulse train in frequency

Example 2.18: Convolve Step and Exponential

� Find y.t/ D Au.t/ � e�˛tu.t/, ˛ > 0

� For t � 0 there is no overlap so Y.t/ D 0

λ0t

No overlap



� For t > 0 there is always overlap

y.t/ D

Z t

0

A � e�˛.t��/ d�

D Ae�˛t �e˛�

˛

ˇ̌̌t0

D Ae�˛t �e˛t � 1

˛

λ0 t

For t > 0 there is always overlap

� Summary,

y.t/ DA

˛

�1 � e�˛t

�u.t/

t

y(t)

A/α

0


CONTENTS

Direct Approach for the FT of a Periodic Signal

� The FT of a periodic signal can be found directly by expandingx.t/ as follows

x.t/ D

"1X

mD�1

ı.t �mTs/

#� p.t/ D

1XmD�1

p.t �mTs/

where p.t/ represents one period of x.t/, having period Ts

� From the convolution theorem

X.f / D F(1X

mD�1

ı.t �mTs/

)� P.f /

D fsP.f /

1XnD�1

ı.f � nfs/

D fs

1XnD�1

P.nfs/ı.f � nfs/

where P.f / D Ffp.t/g

� The FT transform pair just established is

1XmD�1

p.t �mTs/F !

1XnD�1

fsP.nfs/ı.f � nfs/



Example 2.19: p.t/ D ….t=2/C….t=4/, T0 D 10

tT

0 = 100 1-1-2 2

. . .. . .

x(t)

1

2

Stacked pulses periodic signal

� We begin by finding P.f / using Ff….t=�/g D �sinc.f �/

P.f / D 2sinc.2f /C 4sinc.4f /

� Plugging into the FT pair derived above with nfs D n=10,

X.f / D1

10

1XnD�1

�2sinc

�n5

�C 4sinc

�2n

5

��ı�f �

n

10

�

2.5.7 Poisson Sum Formula

� The Poisson sum formula from mathematics can be derivedusing the FT pair

e�j 2�.nfs/tF ! ı.f � nfs/

by writing

F�1(1X

nD�1

fsP.nfs/ı.f � nfs/

)D fs

1XnD�1

P.nfs/ej 2�.nfs/t


CONTENTS

� From the earlier developed FT of periodic signals pair, weknow that the left side of the above is also equal to

1XmD�1

p.t �mTs/alsoD fs

1XnD�1

P.nfs/ej 2�.nfs/t

� We can finally relate this back to the Fourier series coefficients,i.e.,

Xn D fsP.nfs/

2.6 Power Spectral Density and Corre-lation

� For energy signals we have the energy spectral density, G.f /,defined such that

E D

Z1

�1

G.f / df

� For power signals we can define the power spectral density(PSD), S.f / of x.t/ such that

P D

Z1

�1

S.f / df D hjx.t/j2i

– Note: S.f / is real, even and nonnegative

– If x.t/ is periodic S.f / will consist of impulses at theharmonic locations

� For x.t/ D A cos.!0t C �/, intuitively,

S.f / D1

4A2ı.f � f0/C

1

4A2ı.f C f0/


2.6. POWER SPECTRAL DENSITY AND CORRELATION

sinceRS.f / df D A2=2 as expected (power on a per ohm

basis)

� To derive a general formula for the PSD we first need to con-sider the autocorrelation function

2.6.1 The Time Average Autocorrelation Func-tion

� Let �.�/ be the autocorrelation function of an energy signal

�.�/ D F�1˚G.f /

D F�1

˚X.f /X�.f /

D F�1

˚X.f /

� F�1

˚X�.f /

but x.�t /

F ! X�.f / for x.t/ real, so

�.�/ D x.t/ � x.�t / D

Z1

�1

x.t/x.t C �/ d�

or

�.�/ D limT!1

Z T

�T

x.t/x.t C �/ d�

� Observe thatG.f / D F

˚�.�/

� The autocorrelation function (ACF) gives a measure of the

similarity of a signal at time t to that at time t C � ; the co-herence between the signal and the delayed signal


CONTENTS

x(t)

X(f)G(f) = |X(f)|2

φ(τ) =

Energy spectral density and signal relationships

2.6.2 Power Signal Case

For power signals we define the autocorrelation function as

Rx.�/ D hx.t/x.t C �/i

D limT!1

1

2T

Z T

�T

x.t/x.t C �/ dt

if periodicD

1

T0

ZT0

x.t/x.t C �/ dt

� Note that

Rx.0/ D hjx.t/j2i D

Z1

�1

Sx.f / df

and since for energy signals �.�/F ! G.f /, a reasonable

assumption is that

Rx.�/F ! Sx.f /



� A formal statement of this is the Wiener-Kinchine theorem (aproof is given in text Chapter 7)

Sx.f / D

Z1

�1

Rx.�/e�j 2�f � d�

x(t) Rx(τ) S

x(f)

Power spectral density (PSD) and signal relationships

2.6.3 Properties of R.�/

� The following properties hold for the autocorrelation function

1. R.0/ D hjx.t/j2i � jR.�/j for all values of �

2. R.��/ D hx.t/x.t � �/i D R.�/ ) an even function

3. limj� j!1R.�/ D hx.t/i2 if x.t/ is not periodic

4. If x.t/ is periodic, with period T0, thenR.�/ D R.�CT0/

5. FfR.�/g D S.f / � 0 for all values of f

� The power spectrum and autocorrelation function are frequentlyused for systems analysis with random signals

� For the case of random signal, x.t/, the Fourier transformX.f / is also a random signal, but now in the frequency do-main

� The autocorrelation and PSD of x.t/, under typical assump-tions, are both deterministic functions; this turns out to be vitalin problem solving in Comm II and beyond


CONTENTS

Example 2.20: Single Sinusoid

� Consider the signal x.t/ D A cos.2�f0t C �/, for all t

Rx.�/ D1

T0

Z T0

0

A2 cos.2�f0t C �/ cos.2�.t C �/C �/ dt

DA2

2T0

ZT0

�cos.2�f0�/C cos.2�.2f0/t C 2�f0� C 2�/

�dt

DA2

2cos.2�f0�/

� Note that

F˚Rx.�/

D Sx.f / D

A2

4

�ı.f � f0/C ı.f C f0/

�

More Autocorrelation Function Properties

� Suppose that x.t/ has autocorrelation function Rx.�/

� Let y.t/ D AC x.t/, A D constant

Ry.�/ D hŒAC x.t/�ŒAC x.t C �/�i

D hA2i C hAx.t C �/i C hAx.t/i C hx.t/x.t C �/i

D A2 C 2Ahx.t/i„ ƒ‚ …const. terms

CRx.�/

� Let z.t/ D x.t � t0/

Rz.�/ D hz.t/z.t C �/i D hx.t � t0/x.t � t0 C �i

D hx.�/x.�C �/i; with � D t � t0D Rx.�/



� The last result shows us that the autocorrelation function isblind to time offsets

Example 2.21: Sum of Two Sinusoids

� Consider the sum of two sinusoids

y.t/ D x1.t/C x2.t/

where x1.t/ D A1 cos.2�f1tC�1/ and x2.t/ D A2 cos.2�f2tC�2/ and we assume that f1 ¤ f2

� Using the definition

Ry.�/ D hŒx1.t/C x2.t/�Œx1.t C �/x2.t C �/�i

D hx1.t/x1.t C �/i C hx2.t/x2.t C �/i

C hx1.t/x2.t C �/i C hx2.t/x1.t C �/i

� The last two terms are zero since hcos..!1˙!2/t/i D 0 whenf1 ¤ f2 (why?), hence

Ry.�/ D Rx1.�/CRx2.�/; for f1 ¤ f2

DA212

cos.2�f1�/CA222

cos.2�f2�/

Example 2.22: PN Sequences

� In the testing and evaluation of digital communication systemsa source of known digital data (i.e., ‘1’s and ‘0’s) is required(see also text Chapter 10 p. 524–527)


CONTENTS

� A maximal length sequence generator or pseudo-noise (PN)code is often used for this purpose

� Practical implementation of a PN code generator can be ac-complished using an N -stage shift register with appropriateexclusive-or feedback connections

� The sequence length or period of the resulting PN code isM D2N � 1 bits long

CD

1Q

1

CD

2Q

2

CD

3Q

3

M = 23 - 1 = 7

one period = MT

t

x(t)

x(t)

ClockPeriod = T

+A

-A

Three stage PN (m-sequence) generator using logic circuits

� PN sequences have quite a number of properties, one being thatthe time average autocorrelation function is of the form shownbelow



τ

Rx(τ)

T-T

MT

MT

. . .. . .

A2

-A2/M

PN sequence autocorrelation function

� The calculation of the power spectral density will be left as ahomework problem (a specific example is text Example 2.20)

– Hint: To find Sx.f / D FfRx.�/g we useXn

p.t � nTs/F ! fs

Xn

P.nfs/ı.f � nfs/

where Ts DMT

– One period of Rx.�/ is a triangle pulse with a level shift

� Suppose the logic levels are switched from˙A to positive lev-els of say v1 to v2

– Using the additional autocorrelation function propertiesthis can be done

– You need to know that a PN sequence contains one more‘1’ than ‘0’

� Python code for generating PN sequences from 2 to 12 stagesplus 16, is found ssd.py:

def PN_gen(N_bits,m=5):"""Maximal length sequence signal generator.


CONTENTS

Generates a sequence 0/1 bits of N_bit duration. The bits themselves are obtainedfrom an m-sequence of length m. Available m-sequence (PN generators) includem = 2,3,...,12, & 16.

Parameters----------N_bits : the number of bits to generatem : the number of shift registers. 2,3, .., 12, & 16

Returns-------PN : ndarray of the generator output over N_bits

Notes-----The sequence is periodic having period 2**m - 1 (2^m - 1).

Examples-------->>> # A 15 bit period signal nover 50 bits>>> PN = PN_gen(50,4)"""c = m_seq(m)Q = len(c)max_periods = int(np.ceil(N_bits/float(Q)))PN = np.zeros(max_periods*Q)for k in range(max_periods):

PN[k*Q:(k+1)*Q] = cPN = np.resize(PN, (1,N_bits))return PN.flatten()

def m_seq(m):"""Generate an m-sequence ndarray using an all-ones initialization.

Available m-sequence (PN generators) include m = 2,3,...,12, & 16.

Parameters----------m : the number of shift registers. 2,3, .., 12, & 16

Returns-------c : ndarray of one period of the m-sequence

Notes-----The sequence period is 2**m - 1 (2^m - 1).

Examples-------->>> c = m_seq(5)"""# Load shift register with all ones to startsr = np.ones(m)# M-squence length is:Q = 2**m - 1c = np.zeros(Q)



if m == 2:taps = np.array([1, 1, 1])

elif m == 3:taps = np.array([1, 0, 1, 1])

elif m == 4:taps = np.array([1, 0, 0, 1, 1])

elif m == 5:taps = np.array([1, 0, 0, 1, 0, 1])

elif m == 6:taps = np.array([1, 0, 0, 0, 0, 1, 1])

elif m == 7:taps = np.array([1, 0, 0, 0, 1, 0, 0, 1])

elif m == 8:taps = np.array([1, 0, 0, 0, 1, 1, 1, 0, 1])

elif m == 9:taps = np.array([1, 0, 0, 0, 0, 1, 0, 0, 0, 1])

elif m == 10:taps = np.array([1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1])

elif m == 11:taps = np.array([1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1])

elif m == 12:taps = np.array([1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1])

elif m == 16:taps = np.array([1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1])

else:print ’Invalid length specified’

for n in range(Q):tap_xor = 0c[n] = sr[-1]for k in range(1,m):

if taps[k] == 1:tap_xor = np.bitwise_xor(tap_xor,np.bitwise_xor(int(sr[-1]),int(sr[m-1-k])))

sr[1:] = sr[:-1]sr[0] = tap_xor

return c

R.�/, S.f /, and Fourier Series

� For a periodic power signal, x.t/, we can write

x.t/ D

1XnD�1

Xnej 2�.nf0/t

� There is an interesting linkage between the Fourier series rep-resentation of a signal, the power spectrum, and then back tothe autocorrelation function


CONTENTS

� Using the orthogonality properties of the Fourier series expan-sion we can write

R.�/ D

* 1X

nD�1

Xnej 2�.nf0/t

! 1X

mD�1

Xmej 2�.mf0/.tC�/

!�+

D

1XnD�1

1XmD�1

XnX�

m

˝ej 2�.nf0t /e�j 2�.mf0/.tC�/

˛„ ƒ‚ …n¤m terms are zero, why?

D

1XnD�1

jXnj2˝ej 2�.nf0/te�j 2�.nf0/.tC�/

˛D

1XnD�1

jXnj2ej 2�.nf0/�

� The power spectral density can be obtained by Fourier trans-forming both sides of the above

S.f / D

1XnD�1

jXnj2ı.f � nf0/

Example 2.23: PN Sequence Analysis and Simulation

� In this example I consider an N D 4 or M D 15 generatorfrom two points of view:

1. First using the FFT to calculate the Fourier coefficientsXn using one period of the waveform (in discrete-time Iuse 10 samples per bit)

2. Second, I just generate a waveform of 10,000 bits (againat 10 samples per bit) and use standard signal processing



tools to estimate the time average autocorrelation functionand the PSD

� Numerical Fourier Series Analysis: Generate one period ofthe waveform using ssd.m_seq(4) and then upsample and in-terpolate to create a waveform of 10 samples per bit

� The waveform amplitude levels are Œ0; 1� so there is a largeDC component visible in the spectrum (why?); eight ones andseven zeros makes the average value X0 D 8=15 D 0:533

� The function ssd.m_seq() returns an array of zeros and onesof length 15, i.e., len(x_PN4) = 15

� To create a waveform I upsample the signal by 10 and thenfilter using a finite impulse response of exactly 10 ones

� To plot the line spectrum I use the ssd.line_spectra() func-tion used earlier


CONTENTS

15 bit period

DC line 0.533 (-5.46 dB)

f = 1/T = 1 HzLine spacing = 1/(MT) = 1/15 Hz

2sinc ( ) spect. shapex

Fourier series based spectral analysis of PN code



1 115M

- -=

1–1

15 15MT T= =. . . . . .

Same spectral shape and line spacing as direct FS analysis; scaling with PSD function is different however

DC different as waveform values are [-1,1]

Using simulation to estimate Ry.�/ and Sy.f / of a PN code

� In generating an estimate of the autocorrelation function I usethe FFT to find the time averaged autocorrelation function inthe frequency domain

� In generating the spectral estimate, the Python function psd()(frommatplotlib) function uses Welch’s method of averaged peri-odograms 1

� Here the PSD estimate uses a 4096 point FFT and assumedsampling rate of 10 Hz; the spectral resolution is 10/4096 =0.002441 Hz

1http://en.wikipedia.org/wiki/Periodogram


CONTENTS

2.7 Linear Time Invariant (LTI) Systems

x(t) y(t) =

operator

Linear system block diagram

Definition

� Linearity (superposition) holds, that is for input ˛1x1.t/C˛2x2.t/,˛1 and ˛2 constants,

y.t/ D H�˛1x1.t/C ˛2x2.t/

�D ˛1H

�x1.t/

�C ˛2H

�x2.t/

�D ˛1y1.t/C ˛2y2.t/

� A system is time invariant (fixed) if for y.t/ D HŒx.t/�, adelayed input gives a correspondingly delayed output, i.e.,

y.t � t0/ D H�x.t � t0/

�Impulse Response and Superposition Integral

� The impulse response of an LTI system is denoted

h.t/�D H

�ı.t/

�assuming the system is initially at rest

� Suppose we can write x.t/ as

x.t/ D

NXnD1

˛nı.t � tn/


2.7. LINEAR TIME INVARIANT (LTI) SYSTEMS

� For an LTI system with impulse response h. /

y.t/ D

NXnD1

˛nh.t � tn/

� To develop the superposition integral we write

x.t/ D

Z1

�1

x.�/ı.t � �/ d�

' limN!1

NXnD�N

x.n�t/ı.t � n�t/�t; for �t � 1

t

x(t)

0 ∆t−∆t 2∆t 3∆t 4∆t 5∆t 6∆t

Rectangle area is approximation

. . . . . .

Impulse sequence approximation to x.t/

� If we apply H to both sides and let�t ! 0 such that n�t ! �

we have

y.t/ ' limN!1

NXnD�N

x.n�t/h.t � n�t/�t

D

Z1

�1

x.�/h.t � �/ d� D x.t/ � h.t/

orD

Z1

�1

x.t � �/h.�/ d� D h.t/ � x.t/


CONTENTS

2.7.1 Stability

� In signals and systems the concept of bounded-input bounded-output (BIBO) stability is introduced

� Satisfying this definition requires that every bounded-input (jx.t/j <1) produces a bounded output (jy.t/j <1)

� For LTI systems a fundamental theorem states that a system isBIBO stable if and only ifZ

1

�1

jh.t/j dt <1

� Further implications of this will be discussed later

2.7.2 Transfer Function

� The frequency domain result corresponding to the convolutionexpression y.t/ D x.t/ � h.t/ is

Y.f / D X.f /H.f /

where H.f / is known as the transfer function or frequencyresponse of the system having impulse response h.t/

� It immediately follows that

h.t/F ! H.f /

and

y.t/ D F�1˚X.f /H.f /

D

Z1

�1

X.f /H.f /ej 2�f t df



2.7.3 Causality

� A system is causal if the present output relies only on past andpresent inputs, that is the output does not anticipate the input

� The fact that for LTI systems y.t/ D x.t/ � h.t/ implies thatfor a causal system we must have

h.t/ D 0; t < 0

– Having h.t/ nonzero for t < 0 would incorporate futurevalues of the input to form the present value of the output

� Systems that are causal have limitations on their frequency re-sponse, in particular the Paley–Wiener theorem states that forR1

�1jh.t/j2 dt <1, H.f / for a causal system must satisfyZ

1

�1

j ln jH.f /jj1C f 2

df <1

� In simple terms this means:

1. We cannot have jH.f /j D 0 over a finite band of fre-quencies (isolated points ok)

2. The roll-off rate of jH.f /j cannot be too great, e.g., e�k1jf j

and e�k2jf j2

are not allowed, but polynomial forms suchasp1=.1C .f =fc/2N , N an integer, are acceptable

3. Practical filters such as Butterworth, Chebyshev, and el-liptical filters can come close to ideal requirements


CONTENTS

2.7.4 Properties of H.f /

� For h.t/ real it follows that

jH.�f /j D jH.f /j and †H.�f / D �†H.f /

why?

� Input/output relationships for spectral densities are

Gy.f / D jY.f /j2D jX.f /H.f /j2 D jH.f /j2Gx.f /

Sy.f / D jH.f /j2Sx.f / proof in text chap. 6

Example 2.24: RC Lowpass Filter

C

R

x(t)

X(f ) Y(f )

y(t)

h(t), H(f)

ic(t)

vc(t)

RC lowpass filter schematic

� To find H.f / we may solve the circuit using AC steady-stateanalysis

Y.j!/

X.j!/D

1j!c

RC 1j!c

D1

1C j!RC

so

H.f / DY.f /

X.f /D

1

1C jf=f3; where f3 D 1=.2�RC/



� From the circuit differential equation

x.t/ D ic.t/RC y.t/

but

ic.t/ D cdvc.t/

dtD c

y.t/

dt

thus

RCdy.t/

dtC y.t/ D x.t/

� FT both sides using dx=dtF ! j 2�fX.f /

j 2�fRCY.f /C Y.f / D X.f /

so again

H.f / DY.f /

X.f /D

1

1C jf=f3

D1p

1C .f =f3/2e�j tan�1.f =f3/

� The Laplace transform could also be used here, and perhaps ispreferred; we just need to substitute s ! j! ! j 2�f


CONTENTS

f

ff3

-f3

π/2

-π/2

1

RC lowpass frequency response

� Find the system response to

x.t/ D A…

�.t � T=2/

T

�� Finding Y.f / is easy since

Y.f / D X.f /H.f / D AT sinc.f T /�

1

1C jf=fs

�e�j�f t

� To find y.t/ we can IFT the above, use Laplace transforms, orconvolve directly

� From the FT tables we known that

h.t/ D1

RCe�t=.RC/ u.t/



� In Example 2.18 we showed that

Au.t/ � e�˛tu.t/ DA

˛

�1 � e�˛t

�u.t/

� Note that

A…

�t � T=2

T

�D AŒu.t/ � u.t � T /�

and here ˛ D 1=.RC/, so

y.t/ DA

RCRC

�1 � e�t=.RC/

�u.t/

�A

RCRC

�1 � e�.t�T /=.RC/

�u.t � T /

0.5 1 1.5 2 2.5 3

0.2

0.4

0.6

0.8

1

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

t/T fT

fT fT

y(t) |X(f)|, |H(f)|, |Y(f)|

|X(f)|, |H(f)|, |Y(f)| |X(f)|, |H(f)|, |Y(f)|

RC = 2T

RC = T/2RC = T/10

T/10

RC =

T/5

T/2

T

2T

Pulse time response and frequency spectra with A D 1


CONTENTS

2.7.5 Input/Output with Spectral Densities

� We know that for an LTI system with frequency responseH.f /and input Fourier transform X.f /, the output Fourier trans-form is given by Y.f / D H.f /X.f /

� It is easy to show that in terms of energy spectral density

Gy.f / D jH.f /j2Gx.f /

where Gx.f / D jX.f /j2 and Gy.f / D jY.f /j2

� For the case of power signals a similar relationship holds withthe power spectral density (proof found in Chapter text 7, i.e.,Comm II)

Sy.f / D jH.f /j2 Sx.f /

2.7.6 Response to Periodic Inputs

� When the input is periodic we can write

x.t/ D

1XnD�1

Xnej 2�.nf0/t

which implies that

X.f / D

1XnD�1

Xnı.f � nf0/

� It then follows that

Y.f / D

1XnD�1

XnH.nf0/ı.f � nf0/



and

y.t/ D

1XnD�1

XnH.nf0/ej 2�.nf0/t

D

1XnD�1

jXnjjH.nf0jej Œ2�.nf0/tC†XnC†H.nf0/�

� This is a steady-state response calculation, since the analysisassumes that the periodic signal was applied to the system att D �1

2.7.7 Distortionless Transmission

� In the time domain a distortionless system is such that for anyinput x.t/,

y.t/ D H0x.t � t0/

where H0 and t0 are constants

� In the frequency domain the implies a frequency response ofthe form

H.f / D H0e�j 2�f t0;

that is the amplitude response is constant and the phase shift islinear with frequency

� Distortion types:

1. Amplitude response is not constant over a frequency band(interval) of interest$ amplitude distortion

2. Phase response is not linear over a frequency band of in-terest$ phase distortion


CONTENTS

3. The system is non-linear, e.g., y.t/ D k0 C k1x.t/ C

k2x2.t/$ nonlinear distortion

2.7.8 Group and Phase Delay

� The phase distortion of a linear system can be characterizedusing group delay, Tg.f /,

Tg.f / D �1

2�

d�.f /

df

where �.f / is the phase response of an LTI system

� Note that for a distortionless system �.f / D �2�f t0, so

Tg.f / D �1

2�

d

df� 2�f t0 D t0 s;

clearly a constant group delay

� Tg.f / is the delay that two or more frequency componentsundergo in passing through an LTI system

– If say Tg.f1/ ¤ Tg.f2/ and both of these frequencies arein a band of interest, then we know that delay distortionexists

– Having two different frequency components arrive at thesystem output at different times produces signal disper-sion

� An LTI system passing a single frequency component, x.t/ DA cos.2�f1t /, always appears distortionless since at a single



frequency the output is just

y.t/ D AjH.f1/j cos�2�f1t C �.f1/

�D AjH1.f /j cos

�2�f1

�t ��.f1/

2�f1

��which is equivalent to a delay known as the phase delay

Tp.f / D ��.f /

2�f

� The system output now is

y.t/ D AjH.f1/j cos�2�f1.t � Tp.f1//�

� Note that for a distortionless system

Tp.f / D �1

2�f.�2�f t0/ D t0

Example 2.25: Terminated Lossless Transmission Line

x(t) y(t)

Rs = R

0

RL = R

0R

0, v

p

L

y.t/ D1

2x�t �

L

vp

�Lossless transmission line


CONTENTS

� We conclude that H0 D 1=2 and t0 D L=vp

� Note that a real transmission line does have losses that intro-duces dispersion on a wideband signal

Example 2.26: A Fictitious System

-20 -10 10 20

0.5

1

1.5

2

-20 -10 10 20

-1.5

-1

-0.5

0.5

1

1.5

-20 -10 10 20

0.0025

0.005

0.0075

0.01

0.0125

0.015

-20 -10 10 20

0.011

0.012

0.013

0.014

0.015

0.016

f (Hz)

f (Hz)

f (Hz)f (Hz)

H(f)|H(f)|

Tg(f) T

p(f)

No distortion on |f | < 10 Hz band

Ampl. Radians

Time Time

Amplitude, phase, group delay, phase delay

� The system in this example is artificial, but the definitions canbe observed just the same

� For signals with spectral content limited to jf j < 10 Hz thereis no distortion, amplitude or phase/group delay



� For 10 < jf j < 20 amplitude distortion is not present, butphase distortion is

� For jf j > 15 both amplitude and phase distortion are present

� What about the interval 10 < jf j < 15?

2.7.9 Nonlinear Distortion

� In the time domain a nonlinear system may be written as

y.t/ D

1XnD0

anxn.t/

� Specifically consider

y.t/ D a1x.t/C a2x2.t/

� Letx.t/ D A1 cos.!1t /C A2 cos.!2t /

� Expanding the output we have

y.t/ D a1�A1 cos.!1t /C A2 cos.!2t /

�C a1

�A1 cos.!1t /C A2 cos.!2t /

�2D a1

�A1 cos.!1t /C A2 cos.!2t /

�C

na22

�A21 C A

22

�Ca2

2

�A21 cos.2!1t /C A22 cos.2!2t /

�oC a2A1A2

˚cosŒ.!1 C !2/t �C cosŒ.!1 � !2/t �

– The third line is the desired output


CONTENTS

– The fourth line is termed harmonic distortion

– The fifth line is termed intermodulation distortion

f

f f

f

A1

A1

A2

a1A

1

a1A

1

a1A

1A

2

a1A

2a

1a

2A

1

a2A

1

2

2

a2A

1

2

2

a2A

2

2

2a

2(A

1 + A

2)

2 2

2

a2A

1

2

2

2f1

f2

f1

f1

2f1

2f2

f1

f1 f

2-f

1

f1+f

2

f2

0

0

0

0

Non-Linear

Non-Linear

Input

Input

Output

Output

One and two tones in y.t/ D a1x.t/C a2x2.t/ device

� In general if y.t/ D a1x.t/C a2x2.t/ the multiplication theo-rem implies that

Y.f / D a1X.f /C a2X.f / �X.f /

� In particular if X.f / D A…�f=.2W /

�Y.f / D a1A…

�f

2W

�C a22WA

2ƒ

�f

2W



Y( f ) =

=

+f f

f

a1A 2Wa

2A2

Wa2A2

-W

-W

W

W

-2W

-2W

2W

2W

a1A + Wa

2A2

a1A + 2Wa

2A2

Continuous spectrum in y.t/ D a1x.t/C a2x2.t/ device

2.7.10 Ideal Filters

1. Lowpass of bandwidth B

HLP.f / D H0…

�f

2B

�e�j 2�f t0

B B-B -B

H0

slope =-2πt

0

|HLP

(f)| HLP

(f)

f f

2. Highpass with cutoff B

HHP.f / D H0

�1 �….f=.2B//

�e�j 2�f t0

B B-B -B

H0

slope =-2πt

0

|HHP

(f)|

f f

HHP

(f)


CONTENTS

3. Bandpass of bandwidth B

HBP.f / D�Hl.f � f0/CHl.f C f0/

�e�j 2�f t0

where Hl.f / D H0….f=B/

B BH

0

slope = -2πt0

|HBP

(f)| HBP

(f)

f f-f

0f0

-f0

f0

� The impulse response of the lowpass filter is

hLP.t/ D F�1˚H0….f=.2B//e

�j 2�f t0

D 2BH0sincŒ2B.t � t0/�

� Ideal filters are not realizable, but simplify calculations andgive useful performance upper bound results

– Note that hLP.t/ ¤ 0 for t < 0, thus the filter is noncausaland unrealizable

� From the modulation theorem it also follows that

hBP.t/ D 2hl.t � t0/ cosŒ2�f0.t � t0/�D 2BH0sincŒB.t � t0/� cosŒ2�f0.t � t0/�



t t

hLP

(t) hBP

(t)

t0

t0

2BH0 2BH

0

t0 - 1

2B t0 - 1

2Bt0 + 1

2B t0 + 1

2B

Ideal lowpass and bandpass impulse responses

2.7.11 Realizable Filters

� We can approximate ideal filters with realizable filters such asButterworth, Chebyshev, and Bessel, to name a few

� We will only consider the lowpass case since via frequencytransformations we can obtain the others

Butterworth

� A Butterworth filter has a maximally flat (flat in the sense ofderivatives of the amplitude response at dc being zero) pass-band

� In the s-domain (s D �Cj!) the transfer function of a lowpassdesign is

HBU.s/ D!nc

.s � s1/.s � s2/ � � � .s � sn/

where

sk D !c exp��

�1

2C2k � 1

2n

��; k D 1; 2; : : : ; n


CONTENTS

� Note that the poles are located on a semi-circle of radius !c D2�fc, where fc is the 3dB cuttoff frequency of the filter

� The amplitude response of a Butterworth filter is simply

jHBU.f /j D1p

1C .f =fc/2n

Butterworth n D 4 lowpass filter

Chebyshev

� A Chebyshev type I filter (ripple in the passband), is is de-signed to maintain the maximum allowable attenuation in thepassband yet have maximum stopband attenuation

� The amplitude response is given by

jHC.f /j D1p

1C �2C 2n .f /

where

Cn.f / D

(cos.n cos�1.f =fc//; 0 � jf j � fc

cosh.n cosh�1.f =fc//; jf j > fc



� The poles are located on an ellipse as shown below

Chebyshev n D 4 lowpass filter

Bessel

� A Bessel filter is designed to maintain linear phase in the pass-band at the expense of the amplitude response

HBE.f / DKn

Bn.f /

where Bn.f / is a Bessel polynomial of order n (see text) andKn is chosen so that the filter gain is unity at f D 0


CONTENTS

Amplitude Rolloff and Group Delay Comparision

� Compare Butterworth, 0.1 dB ripple Chebyshev, and Bessel

n D 3 Amplitude response

n D 3 Group delay



� Despite DSP being pervasive in communication systems, thereis still a great need for analog filter design and implementationtechnologies

� The table below describes some of the well known constructiontechniques

Filter Construction TechniquesConstructionType

Description of El-ements or Filter

Center Fre-quency Range

Unloaded Q

(typicalFilter Appli-cation

LC (passive) lumped elements DC–300 MHzor higher in in-tegrated form

100 Audio, video,IF and RF

Active R, C , op-amps DC–500 kHzor higher usingWB op-amps

200 Audio and lowRF

Crystal quartz crystal 1kHz – 100MHz

100,000 IF

Ceramic ceramic disks withelectrodes

10kHz – 10.7MHz

1,000 IF

Surface acousticwaves (SAW)

interdigitated fin-gers on a Piezo-electric substrate

10-800 MHz, variable IF and RF

Transmission line quarterwave stubs,open and short ckt

UHF and mi-crowave

1,000 RF

Cavity machined andplated metal

Microwave 10,000 RF

Example 2.27: Use Python to Characterize Standard Filters

� You can use the filter design capability of Python with Scipyor MATLAB with the signal processing to study lowpass, band-pass, bandstop, and highpass filters

� Here I will consider two functions written in Python, one fordigital filters and one for analog filters, to allow plotting of gainin dB, phase in radians, and group delay in samples or seconds


CONTENTS

def freqz_resp(b,a=[1],mode = ’dB’,fs=1.0,Npts = 1024,fsize=(6,4)):"""A method for displaying digital filter frequency response magnitude,phase, and group delay. A plot is produced using matplotlib

freq_resp(self,mode = ’dB’,Npts = 1024)

A method for displaying the filter frequency response magnitude,phase, and group delay. A plot is produced using matplotlib

freqs_resp(b,a=[1],Dmin=1,Dmax=5,mode = ’dB’,Npts = 1024,fsize=(6,4))

b = ndarray of numerator coefficientsa = ndarray of denominator coefficents

Dmin = start frequency as 10**DminDmax = stop frequency as 10**Dmaxmode = display mode: ’dB’ magnitude, ’phase’ in radians, or

’groupdelay_s’ in samples and ’groupdelay_t’ in sec,all versus frequency in Hz

Npts = number of points to plot; defult is 1024fsize = figure size; defult is (6,4) inches

Mark Wickert, January 2015"""f = np.arange(0,Npts)/(2.0*Npts)w,H = signal.freqz(b,a,2*np.pi*f)plt.figure(figsize=fsize)if mode.lower() == ’db’:

plt.plot(f*fs,20*np.log10(np.abs(H)))plt.xlabel(’Frequency (Hz)’)plt.ylabel(’Gain (dB)’)plt.title(’Frequency Response - Magnitude’)

elif mode.lower() == ’phase’:plt.plot(f*fs,np.angle(H))plt.xlabel(’Frequency (Hz)’)plt.ylabel(’Phase (rad)’)plt.title(’Frequency Response - Phase’)

elif (mode.lower() == ’groupdelay_s’) or (mode.lower() == ’groupdelay_t’):"""Notes-----

Since this calculation involves finding the derivative of thephase response, care must be taken at phase wrapping points



and when the phase jumps by +/-pi, which occurs when theamplitude response changes sign. Since the amplitude responseis zero when the sign changes, the jumps do not alter the groupdelay results."""theta = np.unwrap(np.angle(H))# Since theta for an FIR filter is likely to have many pi phase# jumps too, we unwrap a second time 2*theta and divide by 2theta2 = np.unwrap(2*theta)/2.theta_dif = np.diff(theta2)f_diff = np.diff(f)Tg = -np.diff(theta2)/np.diff(w)max_Tg = np.max(Tg)#print(max_Tg)if mode.lower() == ’groupdelay_t’:

max_Tg /= fsplt.plot(f[:-1]*fs,Tg/fs)plt.ylim([0,1.2*max_Tg])

else:plt.plot(f[:-1]*fs,Tg)plt.ylim([0,1.2*max_Tg])

plt.xlabel(’Frequency (Hz)’)if mode.lower() == ’groupdelay_t’:

plt.ylabel(’Group Delay (s)’)else:

plt.ylabel(’Group Delay (samples)’)plt.title(’Frequency Response - Group Delay’)

else:s1 = ’Error, mode must be "dB", "phase, ’s2 = ’"groupdelay_s", or "groupdelay_t"’print(s1 + s2)

def freqs_resp(b,a=[1],Dmin=1,Dmax=5,mode = ’dB’,Npts = 1024,fsize=(6,4)):"""A method for displaying analog filter frequency response magnitude,phase, and group delay. A plot is produced using matplotlib

freqs_resp(b,a=[1],Dmin=1,Dmax=5,mode=’dB’,Npts=1024,fsize=(6,4))

b = ndarray of numerator coefficientsa = ndarray of denominator coefficents

Dmin = start frequency as 10**DminDmax = stop frequency as 10**Dmaxmode = display mode: ’dB’ magnitude, ’phase’ in radians, or

’groupdelay’, all versus log frequency in HzNpts = number of points to plot; defult is 1024


CONTENTS

fsize = figure size; defult is (6,4) inches

Mark Wickert, January 2015"""f = np.logspace(Dmin,Dmax,Npts)w,H = signal.freqs(b,a,2*np.pi*f)plt.figure(figsize=fsize)if mode.lower() == ’db’:

plt.semilogx(f,20*np.log10(np.abs(H)))plt.xlabel(’Frequency (Hz)’)plt.ylabel(’Gain (dB)’)plt.title(’Frequency Response - Magnitude’)

elif mode.lower() == ’phase’:plt.semilogx(f,np.angle(H))plt.xlabel(’Frequency (Hz)’)plt.ylabel(’Phase (rad)’)plt.title(’Frequency Response - Phase’)

elif mode.lower() == ’groupdelay’:"""Notes-----

See freqz_resp() for calculation details."""theta = np.unwrap(np.angle(H))# Since theta for an FIR filter is likely to have many pi phase# jumps too, we unwrap a second time 2*theta and divide by 2theta2 = np.unwrap(2*theta)/2.theta_dif = np.diff(theta2)f_diff = np.diff(f)Tg = -np.diff(theta2)/np.diff(w)max_Tg = np.max(Tg)#print(max_Tg)plt.semilogx(f[:-1],Tg)plt.ylim([0,1.2*max_Tg])plt.xlabel(’Frequency (Hz)’)plt.ylabel(’Group Delay (s)’)plt.title(’Frequency Response - Group Delay’)

else:print(’Error, mode must be "dB" or "phase or "groupdelay"’)



� Case 1: A 5th-order Chebyshev type 1 digital bandpass filter,having 1 dB ripple and passband of Œ250; 300� Hz relative to asampling rate of fs D 1000 Hz

5th-Order-Cheby1 BPFDigitalfs = 1000 Hz

Highly peaked near the band-edge, even with 0.1 dB ripple

Digtal bandpass with fs D 1000Hz (Chebyshev)

� The Chebyshev in both analog and digital forms still has a largepeak in the group delay, even with a small ripple (here 0.1 dB)


CONTENTS

� Case 2: A 7th-order Bessel analog bandpass filter, having pass-band of Œ10; 50� MHz

Not 3dB bandwidth7th-Order Bessel BPFAnalog

Group delay relatively �at in passband, compared with Chebyshev

Analog bandpass (Bessel)

� The Bessel filter has a much lower and smoother group delay,but the magnitude response is rather sloppy

� The filter passband is far from being flat and the roll-off isgradual considering the filter order is seven



2.7.12 Pulse Resolution, Risetime, and Band-width

Problem: Given a non-bandlimited signal, what is a reasonable esti-mate of the signals transmission bandwidth?We would like to obtain a relationship to the signals time duration

� Step 1: We first consider a time domain relationship by seekinga constant T such that

T x.0/ D

Z1

�1

jx.t/j dt

t

x(0)

|x(t)|

T/2-T/2 0

Make areas equal via T

� Note thatZ1

�1

x.t/ dt D

Z1

�1


ˇ̌̌̌fD0

D X.0/

and Z1

�1

jx.t/j dt �

Z1

�1

x.t/ dt

which implies

T x.0/ � X.0/

� Step 2: Find a constant W such that


CONTENTS

2WX.0/ D

Z1

�1

jX.f /j df

f

X(0)

|X(f)|

W-W 0

Make areas equal via W

� Note thatZ1

�1

X.f / df D

Z1

�1

X.f /ej 2�f t df

ˇ̌̌̌tD0

D x.0/

and Z1

�1

jX.f /j df �

Z1

�1

X.f / df

which implies that

2WX.0/ � x.0/

� Combining the results of Step 1 and Step 2, we have

2WX.0/ � x.0/ �1

TX.0/

or

2W �1

Tor W �

1

2T

Example 2.28: Rectangle Pulse

� Consider the pulse x.t/ D ….t=T /

� We know that X.f / D T sinc.f T /



-1 -0.5 0.5 1

0.2

0.4

0.6

0.8

1

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

fTt /T

x(t) |X(f)|/T

f1/(2T)-1/(2T)

Lower bound for W

Pulse width versus Bandwidth, is W � 1=.2T ?

� We see that for the case of the sinc. / function the bandwidth,W , is clearly greater than the simple bound predicts

Risetime

� There is also a relationship between the risetime of a pulse-likesignal and bandwidth

� Definition: The risetime, TR, is the time required for the lead-ing edge of a pulse to go from 10% to 90% of its final value

� Given the impulse response h.t/ for an LTI system, the stepresponse is just

ys.t/ D

Z1

�1

h.�/u.t � �/ d�

D

Z t

�1

h.�/ d�if causalD

Z t

0

h.�/ d�

Example 2.29: Risetime of RC Lowpass


CONTENTS

� The RC lowpass filter has impulse response

h.t/ D1

RCe�t=.RC/u.t/

� The step response is

ys.t/ D�1 � e�t=.RC/

�u.t/

� The risetime can be obtained by setting ys.t1/ D 0:1 and ys.t2/ D0:9

0:1 D�1 � e�t1=.RC/

�) ln.0:9/ D

�t1

RC

0:9 D�1 � e�t2=.RC/

�) ln.0:1/ D

�t2

RC

� The difference t2 � t1 is the risetime

TR D t2 � t1 D RC ln.0:9=0:1/ ' 2:2RC D0:35

f3

where f3 is the RC lowpass 3dB frequency

Example 2.30: Risetime of Ideal Lowpass

� The risetime of an ideal lowpass filter is of interest since it isused in modeling and also to see what an ideal filter does to astep input

� The impulse response is

h.t/ D F�1�…

�f

2B

��D 2BsincŒ2Bt�



� The step response then is

ys D

Z t

�1

2BsincŒ2B�� d�

D1

�

Z 2�Bt

�1

sinuu

du

D1

2C1

�SiŒ2�Bt�

where Si( ) is a special function known as the sine integral

� We can numerically find the risetime to be

TR '0:44

B

1 2 3 4 5

0.2

0.4

0.6

0.8

1

-2 -1 1 2 3

0.2

0.4

0.6

0.8

1

t RC t/B

Step Response of RC Lowpass Step Response of Ideal Lowpass

2.2 0.44

RC and ideal lowpass risetime comparison


CONTENTS

2.8 Sampling Theory

Integrate with Chapter 3 material.

2.9 The Hilbert Transform

Integrate with Chapter 3 material.

2.10 The Discrete Fourier Transform andFFT

?


2.10. THE DISCRETE FOURIER TRANSFORM AND FFT

.


signal and linear system analysis and linear system analysis contents 2.1 signal models ... a signal...

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