Handout FS

8/11/2019 Handout FS

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Analog and Digital Communications (Lecture notes on basic signal

properties and Fourier series)

Dr. Laxminarayana S Pillutla

DA-IICT, Gandhinagar, Gujarat, India.

Email: laxminarayana [email protected]

Signals and Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Energy and Power definitions (ctd...) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Types of Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Special Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Special Signals (ctd...). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Inner product operator and orthogonality of signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Orthogonality of Complex Exponential Signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Orthogonality of Complex Exponential Signals (ctd...) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Orthogonality of Complex Exponential Signals (ctd...) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Exponential Fourier Series for Periodic Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Generalization of Exponential Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Generalization of Exponential Fourier Series (ctd...). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Properties of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Properties of Fourier Series (ctd...) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Trigonometric Fourier Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Parsevals Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Example on Exponential FS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Example on Exponential FS (ctd...) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Example on Exponential FS (ctd...) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Gibbs Phenomenon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Gibbs Phenomenon (ctd...). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Gibbs Phenomenon (ctd...). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Gibbs Phenomenon Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Dirichlets conditions for Convergence of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Dirichlets conditions for Convergence of Fourier Series (ctd...) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Dirichlets conditions for Convergence of Fourier Series (ctd...) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

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Signals and Systems A signal is a set of information or data. Examples: (a) voice signal (b) stock market data, etc.

A signal is mathematically represented as a function of independent variable say time, space, etc.

For example a speech signal is usually a function of time, while the charge density is a function of space.

A system synthesizes/processes information contained in a signal. Examples: Radar, communication

channel, etc. Signal Quantification:A signal is typically measured in terms of two quantities namely: energy and power.

Energyof a real/complex signal is defined as

Eg

+

g2(t)dt(real signal) (1)

+

|g(t)|2dt(complex signal) (2)

cDr. Laxminarayana S Pillutla CT 214 - Analog and Digital Communications 2 / 27

Energy and Power definitions (ctd...) Comments on Energy definition:

For Eg to be meaningful Eg


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Types of Signals Continuous timesignals are those that exist for every t R, where R is the set of real numbers. Example:

Any continuous function of timet.

Discrete timesignals are those that exist fort D, where D is some discrete set. Example: Stock marketdata, sampled version of a continuous time signal, etc.

Periodic and aperiodic signalsare those that repeat after every time period T0. Aperiodic signals are those

that dont repeat after certain time. Energy type signalsare those for which the energyEsatisfies the conditionE


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Inner product operator and orthogonality of signals

Inner product of signalsg(t)andh(t): denoted as< g(t), h(t)> is defined as follows:

Energy type signals: Ifg(t)andh(t)are complex energy type signals then

< g(t), h(t)> +

g(t)h(t)dt (8)

(Note: EnergyEg of an energy type signal g(t)is equal to< g(t), g(t)>) Power type signals: Ifg(t)andh(t)are complex power type signals then

< g(t), h(t)> limT

1

T

+T/2T/2

g(t)h(t)dt (9)

(Note: PowerPg of a power type signal g(t)is equal to< g(t), g(t)>)

Two signalsg(t)andh(t)(of energy/power type) are said to beorthogonalif their inner product< g(t), h(t)> is equal to zero.


Orthogonality of Complex Exponential Signals

Consider two complex exponential functionsej1t andej2t at different angular frequencies say1 and2.

Infinite duration case wheret (, +): Since a complex exponential is power type signal,therefore,

< ej1t, ej2t > = limT

1

T +T/2

T/2ej1tej2tdt

...

= limT

2

T(1 2)sin

(1 2)T

2

= 0(| sin

(1 2)T

2

| 1, T) (10)

Therefore complex exponentials of infinite duration are always orthogonal, irrespective of 1and2.


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Orthogonality of Complex Exponential Signals (ctd...)

Orthogonality for the finite duration case. Here we assumeTas the fundamental period of the complexexponential signal.

Finite duration case:Here we assume the exponential signals to be of finite duration. Taking the inner

product of the two complex exponential signals over one fundamental period we obtain

< ej1t, ej2t > = 1

T

T/2T/2

ej1tej2tdt

...

= 2

T

sin(12)T

2

(1 2)

= 0 (for1=2k1

T and2=

2k2T

),

wherek1 andk2 are integers.


Orthogonality of Complex Exponential Signals (ctd...)

Thus, for the finite duration case orthogonality of complex exponentials holds only when the frequencies are

integer multiples of the fundamental period.

Significance:

The orthogonality property and the eigenfunction function property of complex exponential functions are

the primary reasons for their usage in communication systems.

Indeed one of the latest multiplexing technique called orthogonal frequency division multiplexing (OFDM)

exploits the above two properties.


Exponential Fourier Series for Periodic Signals

A periodic signal with finite power over one period can be expressed in the form ofexponential Fourier series.

Ifg(t)is a periodic signal of periodT0 seconds, then

g(t) =+

n=Dnej0t , (11)whereDn =

1

T0

g(t)ejn0tdt (12)

Note:0=2

T0(13)

In general sinceDsn are complex, thereforeDn = |Dn|e

jDn .

To specify the spectra of a signalg(t)we need two plots: the amplitude spectrum (|Dn| versusn; this yieldsaline spectrum) and the phase spectrum (Dn versusn).


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Generalization of Exponential Fourier Series In general for any power type signalg(t)such thatPg =

1 m= n0 m=n

(15)

The weightsgn, (n= N + N)are determined such that the power Pe of the error signale(t) =g(t) g(t)is minimized. i.e., determinegn so that

Pe limT

1

T

T/2T/2

|e(t)|2dt , (16)

is minimized.


Generalization of Exponential Fourier Series (ctd...)

Indeed it turns out that the coefficientsgn to minimize(17)are given by

gn =< g(t), n(t)> . (17)

AsN ,g(t) g(t)in the sense thatPe 0.

For the special case of exponential FS we choose the orthonormal signal set {n(t), n= N , +N}as

ejn0t , n= N , +N

, which are indeed orthogonal (as long asn are integers) as discussedbefore and also have unit power.

We shall revisit this point while discussing the convergence of exponential FS.


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Properties of Fourier Series

Spectral Properties of FS:

The two-sided line spectrum has a uniform spacing. Each spectral component is separated byf0= 1T0

.

D0 = 1T0

|g(t)|dtgives the average value of the signal g(t). Conjugate symmetry property of real signals: Ifg(t)is real, then

Dn = 1

T0

g(t)ejn0tdt

= 1

T0

g(t)ejn0t

dt(g(t)is real)

=

1

T0

g(t)ejn0tdt

= Dn

orDn = D

n (18)


Properties of Fourier Series (ctd...) Spectral Properties of FS (ctd...):

The property in(18)implies

|Dn| = |Dn|(|D

n|= |Dn|)

Dn = Dn

Thus for real signals, the magnitude spectrum has even symmetry, while the phase spectrum has odd

symmetry. Using the conjugate symmetry property of real signals, the exponential FS can be further simplified into

what is known as trigonometric Fourier series (as shown next).


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Example on Exponential FS

We are interested in the computation of exponential FS of a periodic rectangular pulse train shown in Fig. 2:

T0

+ 2

T0+T0

2

2

T0

2 0

A

v(t)

Figure 2: The figure above shows a periodic rectangular pulse train of ON duration seconds.

The functional form of the rectangular pulse of duration isAt

, where

t

1 |t|< 20 otherwise

(22)

c

Dr. Laxminarayana S Pillutla CT 214 - Analog and Digital Communications 18 / 27

Example on Exponential FS (ctd...) Since the rectangular pulse trainv(t)in Fig. 2 is real and periodic, therefore, we can express it using

Trigonometric FS as

v(t) = D0+ 2n=1

|Dn| cos(n0t + Dn) (23)

whereDn = 1

T0

+T0/2T0/2

v(t)ejn0tdt (24)

= 1T0

+/2/2

Aejn0tdt

= A

T0

+/2/2

Aejn0tdt

...

= Af0sinc (nf0) , (f0 1

T0) (25)

(Note: sinc(x) sin(x)x ).


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Example on Exponential FS (ctd...) Note:D

sn in (25) can be written using Sinc(x)definition as follows:

Dn = A

nsin(nf0) (26)

|Dn| = A

nsin(nf0) (27)

Note:D0 = Af0 (28)

Using(27)and (28) we can simplify the trigonometric FS ofv(t)in (24) as

v(t) =Af0++n=1

2A

nsin(nf0)cos(n0t)( Dn= 0) (29)


Gibbs Phenomenon

The Fig.3shows one period of the periodic rectangular pulse train and its reconstruction via Fourier seriesusingN= 10harmonics:

5 4 3 2 1 0 1 2 3 4 50.2

0

0.2

0.4

0.6

0.8

1

1.2

"Gibbs ears"

Figure 3: The figure above shows the original rectangular pulse and its reconstructed version using Fourier

series with10harmonics. In the reconstructed signal, at the points of discontinuity one can observe oscillatorybehaviour.


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Gibbs Phenomenon (ctd...) The Fig.4shows one period of the periodic rectangular pulse train and its reconstruction via Fourier series

usingN= 100harmonics:

5 4 3 2 1 0 1 2 3 4 50.2

0

0.2

0.4

0.6

0.8

1

1.2

"Gibbs ears"


series with100 harmonics. With the increase in number of harmonicsN to 100 the oscillatory behaviour inthe reconstructed signal has somewhat subsided.


Gibbs Phenomenon (ctd...) The Fig.5shows one period of the periodic rectangular pulse train and its reconstruction via Fourier series

usingN= 1000 harmonics:

5 4 3 2 1 0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


series with1000 harmonics. With the increase in number of harmonics to 1000 the reconstructed signal hassmoothed out considerably.


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Gibbs Phenomenon Summary

The ringing effect observed in Figures3, 4and5is known as Gibbs phenomenon. It occurs in signals at the

points of discontinuity.

As was seen the ringing is pronounced when the number of harmonics are small.

At the points of discontinuity, the Fourier series converges to the average of left and right limit values i.e., ifx0

is the point of discontinuity then the Fourier series converges to

1

2 f(x0) + f(x+0). For example in thecase ofv(t)in Fig. 2,it would converge toA/2at the points of discontinuity. At all the other points where the signal is continuous the FS indeed converges to the value of signal at that

point.


Dirichlets conditions for Convergence of Fourier Series

Dirichlet providedsufficient conditionsfor the convergence of FS of a periodic signal (say)g(t), which aregiven as follows:

The periodic signalg(t)should be absolutely integrable over one period of T0 seconds i.e.,

|g(t)|< (30)

The periodic signalg(t)should have finite number of maxima/minima and discontinuities over one periodofT0 seconds.

Note:As noted above Dirichlets conditions are only sufficient and not necessary, which means if a signal

violates Dirichlets conditions then its FS may/may not converge.


Dirichlets conditions for Convergence of Fourier Series (ctd...)

Under Dirichlets conditions FS converges to the signal value at that point uniformly, provided the signal is

continuous at that point.

Uniform convergence: In general, we say that the sequence of functions {fn} converge uniformly to alimiting functionf, if for every >0, there exists a natural number Nsuch that for alln N,|fn(x) f(x)|< .

In lieu of the above definition one can elaborate on the uniform convergence of FS of a periodic signalg(t).To this end, let us define gK(t) =

+Kn=KDne

jn0t. Then under Dirichlets conditions for every >0, it is

possible to find aNsuch that for allKNand allt

|gK(t) g(t)|< (31)


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Dirichlets conditions for Convergence of Fourier Series (ctd...) Since Dirichlets conditions are only sufficient conditions for the convergence of FS, therefore one can give

alternate condition(s) to that of Dirichlet, that would still ensure the convergence of FS. One such condition is

the finite power per period condition, which ensures convergence in the mean squared error sense

Mean squared error convergence of FS:Letg(t)denote a periodic signal such that its average power overone period is finite. Further, as before, denote gK(t)

+Kn=KDne

jn0t denote the(2K+ 1) harmonic

approximation of the signalg(t), then

limK

| g(t) gK(t) =error

|2dt 0 (32)


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