signal and system analysis objectives: to define the energy and power of a signal to classify...

Signal and System AnalysisSignal and System Analysis

Objectives:•To define the energy and power of a signal•To classify different types of signals•To introduce special functions commonly used in telecom.•To define linear and time-invariant systems•To define convolution•To introduce Fourier series and Fourier transform•To explain the concept of negative frequency•To show how the signal may be described in either the time domain or the frequency domain and establish their relationship.•To study autocorrelation and power spectral density

Classification of SignalsClassification of Signals

As mentioned before, a signal represents the message that is to be sent across the channel. Let’s start looking into more detail.

Signals can be classified in various ways:•Continuous- or discrete-time signals

Signals associated with a computer are discrete-time signals

t

g(t)

-10 -8 -6 -4 -2 0 2 4 6 8 10

ng

[n]


Periodic and nonperiodic signalsA periodic signal is one that repeats itself exactly after a fixed length of time. g(t) = g(t + T) for all t g[n] = g[n+N] for all n

The smallest positive number T (or N) that satisfies the above equation is called the period.

0 1 2 3 4 5 6 7 8 9 10

ng

[n]

N


Deterministic and random signalsDeterministic signal: can be mathematically characterized completely in the time domain.

Random signal: specified only in terms of probabilistic description

All signals encountered in telecommunications are random signals. If a message is used to convey information, it must have some uncertainty (randomness) about it. Otherwise, why communicate?

)cos()( ttg c

22 )]([ ,0)]([ ),( tnEtnEtn

)cos(][ nng c

• We know ,power • Energy • For our purpose, we will neglect the• Energy and average power of a signal and

Power and energy of a signalPower and energy of a signal

IRV RIRVP 22 /

R

T

TT

dttgT

P2

)(2

1lim

)(tg

PdtE

][ng

N

NNng

NP

2][

12

1lim

dttgE2

)(

2

][ngE

Some important signalsSome important signals

Singularity functions in continuous-time systems•Singularity functions are discontinuous or have discontinuous derivatives. •Singularity functions are mathematical idealizations and, strictly speaking, do not occur in physical systems. They serve as good approximations to certain limiting conditions in physical systems.

We will discuss two types of singularity functions: unit step function and

unit impulse function

Step functionStep function

Unit step functionThe unit step function is defined as

1 0

0 0( ) {

t

tu t

(u(t) has no definition at t = 0, or one may define u(0) = 1 or u(0) = ½)

00

01][

n

nnu

-5 -4 -3 -2 -1 0 1 2 3 4 5

1

n

u[n]

Impulse functionImpulse function

Unit impulse function The unit impulse function is defined as

( ) 0, 0

( ) 1

t t

t dt

0

( ) ( )2 2( ) lim

u t u tt

-5 -4 -3 -2 -1 0 1 2 3 4 5

1

[n]

00

01][

n

nn

Relationship between Relationship between (t) and u(t)

( ) ( )

( )( )

td u t

du tt

dt

][00

01][ nu

n

nk

n

k

][]1[][ nnunu

-5 -4 -3 -2 -1 0 1 2 3 4 5

1

n

u[n]

-5 -4 -3 -2 -1 0 1 2 3 4 5

1

[n]

Multiplication of a function by (t)f(t)(t) = f(0)(t), f(t) continuous at t=0.f(t)(t – T) = f(T)(t – T), f(t) continuous at t=T.

Sampling property of (t), [n]

( ) ( ) (0) ( ) (0)

( ) ( ) ( )

f t t dt f t dt f

f t t T dt f T

-1 -0.5 0 0.5 1

1

(t)

f(t)

Further properties of Further properties of (t), [n]

][][][

]0[][][

NfNnnf

fnnf

Exponential functionsExponential functions

tjetg )(

tjejdt

tdg )(

njeng ][

)sin()cos( tjte tj Euler’s formula

Exponential functions are important class of functions in this course. We will be making use of this fact a lot in this course

SystemsSystems

A system is defined as a set of rules that maps an input signal to an output signal.

g(t) -- input signal (or source signal); y(t) -- output signal (or response signal);

Input and response are represented as g(t) y(t) and read as input g(t) causes a response y(t).

Some properties of systemsSome properties of systems

Linear and nonlinear systemsFor a system with g1(t) y1(t) and g2(t) y2(t),A system is said to be linear if the following properties hold:

1) ag1(t) ay1(t) 2) ag1(t) + bg2(t) ay1(t) +by2(t) for any scalar a,b

Otherwise, the system is nonlinear. Examples

)(2)( ttgty )()(2)( 2 tgtgty

Time-invariant and time-varying systemsA system is time-invariant if a time shift in the input results in a corresponding time shift in the output

g(t – t0) y(t – t0) for any t0.i.e. when the same input is applied to a system today or tomorrow, the output is the same, just shifted in time accordingly

Some properties of systemsSome properties of systems

t

t

t

t0t 0t

Exercise

Any system not meeting this requirement is said to be time-varying.

Example:

Another example: Humans

We will focus on Linear and Time-Invariant (LTI) systems inthis course

Time InvarianceTime Invariance

)(2)( ttgty

)()(2)()(2)( 0000 ttgttttytttgty

ExerciseExercise

2)()( tgty

))(log()( tgty

][2)1(][ ngnny

]1[][][ ngngny

Non-linear, time-invariant

)cos()()( ttgty


Linear, time-variant


Linear, time-invariant

Convolution in LTI systemsConvolution in LTI systems

Consider a discrete-time LTI system. If we apply the impulse function as the input, let be the output. ][n ][nh

We call the Impulse response of an LTI system ][nhNow, since the system is linear, if we apply a scaled version ofthe impulse ,

n

1][n

n

][nh

0 0

][nb

n

b

0 n

][nbh

0

b][nb


Furthermore, since the system is time-invariant, if we apply a delayed version of the impulse ,

][ kn

n

1][ kn

n

][ knh

k k

Now, what if we have this?

n n

][][][][ 2121 nbynaynbgnag

0 1 2 0 1 2 3 4 5


n n0 1 2 0 1 2 3 4 5

]1[]2[]2[]1[]3[]0[]3[ hghghgy

][ny][ng

]2[]2[]1[]1[][]0[][ ngngngng If we define]0[]0[]0[ hgy

]0[]1[]1[]0[]1[ hghgy

]0[]2[]1[]1[]2[]0[]2[ hghghgy

n

][nh

0

Do you see a pattern?

k

knhkgny ][][][

In an LTI system with input and impulse response ,][ng ][nh

Similarly for continuous-time systems,

dthgty )()()(


)()()( thtgty

We write convolution as

][][][ nhngny

Example:

n0

1

1 2 n0

1

1 2

n0

1

1 2 3 4

2

3


In an LTI system, the impulse response or completelycharacterize the system.

][ny][nh )(th][ng

LTI system

][nh )(th

)()()( thtgty ][][][ nhngny

Fourier AnalysisFourier Analysis

Fourier analysisAlternative Representation of a periodic signal

t

Ttg

2cos)(

A signal can be represented in either the time domain (where it is a waveform as a function of time) or in the frequency domain. Such representation is called the spectrum of the original time-domain signal.If the signal is specified in the time domain, we can determine its spectrum, and vice versa.Fourier analysis provides a link between the time domain and the frequency domain.

t

Tt

Ttg

22cos5.0

2cos)(

T

2

1

T

22

5.0

Amplitude

T

2

1Amplitude

Fourier AnalysisFourier Analysis

Claim: A periodic function g(t) of period T, can be expressed as an infinite sum of sinusoidal waveforms with frequency This summation is called Fourier series. Fourier series can be written in several forms. One such form is the trigonometric Fourier series:

,...2

3,2

2,2

TTT

11

0

1

0

0

2sin

2cos

2

2sinsin

2coscos

2

2cos)(

nn

nn

nnn

nn

nnn

tT

nbtT

naa

tT

nAtT

nAa

tT

nAtg

The constants are called Fourier Coefficientsnn ba ,

/ 2

/ 2

/ 2

/ 2

2 2( )cos( ) , 0, 1, 2,

2 2( )sin( ) , 1, 2,

n

n

T

T

T

T

n ta g t dt n

T T

n tb g t dt n

T T

Obtaining the Fourier CoefficientsObtaining the Fourier Coefficients

11

0 2sin

2cos

2)(

nn

nn t

Tnbt

Tna

atg

2/

2cos

2sin

2cos

2cos

2cos

2

2cos)(

2/

2/1

1

2/

2/

2/

2/

02/

2/

k

T

Tn

n

n

T

Tn

T

T

T

T

Ta

dttT

ktT

nb

dttT

ktT

na

dttT

ka

dttT

ktg

Fundamental Frequency and Fundamental Frequency and HarmonicsHarmonics

Given with period T, in the formulation

11

0 2sin

2cos

2)(

nn

nn t

Tnbt

Tna

atg

)(tg

is referred to as the fundamental frequency and the others are called harmonics of g(t)

tT

btT

a 2

sin2

cos 11

ExampleExample

)(tg

n

Tndt

T

nt

Tdt

T

nttg

Ta

T

Tn

)/sin(22cos

22cos)(

2 2/

2/

2/

2/

02

sin22

sin)(2 2/

2/

2/

2/

dt

T

nt

Tdt

T

nttg

Tb

T

Tn

1

Fourier serious approximation to periodic Fourier serious approximation to periodic functionsfunctions

1

0 2cos

2)(

nn t

Tna

atg

n

Tna

Ta n

)/sin(2,

20

In practice,

N

nn t

Tna

atg

1

0 2cos

2)(

As , the Fourier series approaches the original function N

Sinc functionSinc function

Rewrite:

If we define a function

and define sinc(0)=1, we may rewrite

n

Tna

Ta n

)/sin(2,

20

Tn

Tn

Tan /

)/sin(2

x

xx

)sin(

)(csin

)/(csin2

TnT

an

This is called the sinc function

Sinc functionSinc function

Exponential Fourier seriesExponential Fourier series

This is called the exponential Fourier series

0( ) jn tn

n

g t c e

0

/ 2

/ 2

1( ) , 0, 1, 2,

Tjn t

n

T

c g t e dt nT

10

10

0

11

0

sincos2

2sin

2cos

2)(

nn

nn

nn

nn

tnbtnaa

tT

nbtT

naa

tg

Since and we can rewrite aswhere

0 = 2/T

Exponential Fourier seriesHow are the coefficients in the exponential Fourier series cn related to an , bn?

0 00 0

1 1 1 1

0 0

1 1

01

2 2( ) cos( ) sin( ) cos( ) sin( )

2 2

[ ( ) ( )] ( )2 2 2 2 2 2

(

o o o o o o

o

n n n nn n n n

jn t jn t jn t jn t jn t jn tn n n n n n

n n

jn t jnn n

n

a an t n tg t a b a n t b n t

T T

a a b a a jb a jbe e e e e e

j

c c e c e

)ot

where c0 = a0 / 2, cn = (an – jbn) / 2, c-n= (an + jbn) / 2

Exponential Fourier seriesExponential Fourier series

Fourier coefficients Fourier coefficients cn in complex fourier ser in complex fourier seriesies

cn can be complex.

•| cn | is called spectral amplitude and represents the amplitude of nth harmonic. Arg(cn) is known as the spectral phase

•Graphic representation of spectral amplitude along with the spectral phase is called complex frequency spectrum of the original signal g(t)

Since c0 = a0 / 2, cn = (an – jbn) / 2, c-n= (an + jbn) / 2,

n

tjnn

nn

nn ectnbtna

atg 0

10

10

0 sincos2

)(

Negative Frequency ~!?!?Negative Frequency ~!?!?

Since n takes on negative values, apprently there is “negative frequency”. But remember, this exists only because we want to express and as

In reality, frequencies can only have positive values.However, it is mathematically easier to use exponential representation rather than trigonometric. That’s why we allow the existence of negative frequency. Keep this in mind ~

tn 0cos tn 0sin

A bit of revisionA bit of revision

Up to now, we have shown that a periodic function in time g(t) canbe specified in two equivalent ways:Time domain representation -- waveform.Frequency domain representation – spectrum, Fourier coefficients.

If the signal is specified in time domain, we can determine itsspectrum and vice versa.

n

tjnn

nn

nn ectnbtna

atg 0

10

10

0 sincos2

)(

0

/ 2

/ 2

1( ) , 0, 1, 2,

Tjn t

n

T

c g t e dt nT

Spectrum dependence on period of Spectrum dependence on period of signalsignal

When T is larger, becomes smaller, the spectrum becomes denser.

When T goes to infinity, • Only a single pulse of width in the time domain. 0 0, i.e., no spacing is left between two line-components; Thus, the spectrum becomes continuous and exists at all frequencies.

(However, there is no change in the shape of the envelope of the spectrum).

T/20

Spectrum dependence on period of Spectrum dependence on period of signalsignal

Fourier TransformFourier Transform

The Fourier transform of a signal g(t) is defined by

and g(t) is called the inverse Fourier transform of G()

( ) ( ) j tG g t e dt

1( ) ( )

2j tg t G e d

The functions g(t) and G() constitute a Fourier transform pair: g(t) G()

G() = F[g(t)]and g(t) = F -1[G()]

What is the difference between Fourier transform and Fourier series?

Fourier TransformFourier Transform

Fourier transform is different from the Fourier Series in that its frequency spectrum is continuous rather than discrete.

Fourier transform is obtained from Fourier series by letting T (for a nonperiodic signal).

The original time function can be uniquely recovered from its Fourier transform.

Fourier Transform and Fourier SeriesFourier Transform and Fourier Series

Please keep in mind that

• A periodic signal spectrum has finite amplitudes and exists at discrete set of frequencies. Those amplitudes are also called the Fourier coefficients of the periodic signal

• A non-periodic signal has a continuous spectrum G() and exist at all frequencies.

Fourier transform of some useful functions

Rectangular function:

Proof

12 2( )

0

( ) ( )2

{tt

rectelsewhere

trect Sa

/ 2 / 2 / 2 / 2 / 2/ 2

/ 2/ 2

2( ) sin ( )

2 2

j t j j j jj t e e e e e

G e dt Saj j j

Unit impulse function: (t) 1 and 1 2()

Proof 0

1

[ ( )] ( ) ( ) ( ) 1

1 1[ ( )] ( )

2 2

j t

j t

F t t e dt t e dt t dt

F e d


Sinusoidal function cos(0t)cos(0t) [( + 0) + ( - 0)]

0

0 0

10 0

0 0

0 0 0

1 1[ ( )] ( )

2 2

2 ( ), 2 ( )

cos [ ( ) ( )]

j tj t

j t j t

F e d e

e e

t

Proof


Properties of Fourier TransformProperties of Fourier Transform

Linearity propertyIf g1(t) G1() and g2(t) G2()then a1g1(t) + a2g2(t) a1G1() + a2G2()where a1 and a2 are constants

This property is proved easily by linearity property of integrals used in defining Fourier transform

Symmetry propertyIf g(t) G(), then G(t) 2g(- )

Proof1

( ) ( )2

2 ( ) ( )

j t

j t

g t G e d

g t G e d

2 ( ) ( )

( ) 2 ( )

j tg G t e dt

G t g

we can interchange the variable t and , i.e. let t , t, then


Time scaling property 1

( ) ( )g at Ga a

[ ( )] ( ) j tF g at g at e dt

let x = at, then dt = dx/a, case 1: when a > 0,

/1 1[ ( )] ( ) ( )j x aF g at g x e dx G

a a a

case 2: when a < 0, then t leads to x - ,/ /1 1 1

[ ( )] ( ) ( ) ( )j x a j x aF g at g x e dx g x e dx Ga a a a

Combined, the two cases are expressed as, 1( ) ( )g at G

a a

Proof


Important Observation:Time domain compression of a signal results in spectral expansionTime domain expansion of a signal results in spectral compression


Time shifting property0

0( ) ( ) j tg t t G e

0 0[ ( )] ( ) j tF g t t g t t e dt

put t – t0 = x, so that dt = dx, then

0 0 0( )0[ ( )] ( ) ( ) ( )j x t j t j tj xF g t t g x e dx e g x e dx G e

Proof

Proof

00( ) ( )j tg t e G

Frequency shifting property

0 0 0( )0[ ( ) ] ( ) ( ) ( )j t j t j tj tF g t e g t e e dt g t e dt G


Significance• Multiplication of a function g(t) by exp(j0t) is equivalent to shifti

ng its Fourier transform in the positive direction by an amount 0. -- Frequency translation theorem.

• Translation of a spectrum helps in achieving modulation, which is performed by multiplying the known signal g(t) by a sinusoidal signal.

0 00

1( )cos [ ( ) ( ) ]

2j t j tg t t g t e g t e

0 0 0

1( )cos [ ( ) ( )]

2g t t G G

Therefore,


• The multiplication of a time function with a sinusoidal function translates the whole spectrum G() to 0.

• exp(j0t) can also provide frequency translation, but it is not a real signal. Hence, sinusoidal function is used in practical modulation system.

Modulation TheoremModulation Theorem

ConvolutionSuppose that g1(t) G1() and g2(t) G2(), then,what is the waveform of g(t) whose Fourier transform is the pro

duct of G1() and G2()? This question arises frequently in spectral analysis, and is ans

wered by the convolution theorem.

The convolution of two time function g1(t) and g2(t), is defined by the following integral

1 2 1 2( ) ( ) ( ) ( )g t g t g g t d


Convolution TheoremConvolution Theorem

Time convolution theoremIf g1(t) G1() and g2(t) G2()Then g1(t) * g2(t) G1()G2()

1 2 1 2

( )1 2 1 2

1 2

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

( )[ ( ) ] ( ) ( )

( ) ( )

j t

j t j j

F g t g t g g t d e dt

g g t e dt e d g G e d

G G

Frequency convolution theoremIf g1(t) G1() and g2(t) G2()

Then 1 2 1 2

1( ) ( ) ( ) ( )

2g t g t G G

The proof is similar to time convolution theorem.

Convolution Theorem: ApplicationsConvolution Theorem: Applications

g1(t) * g2(t) G1()G2()

)()( 02 tttg 0)()()( 101tjeGtttg

)()( 0110 ttgeG tj

If we let , then

But (time shifting property)

Therefore, convolving with a delta function shifted in time by corresponds to a shift of the original signal by

0t0t

)(1 tg

Signal transmission through a linear Signal transmission through a linear systemsystem

y(t) = g(t) * h(t)when g(t) G(), h(t) H(), y(t) Y(), h(t) is the impulse response, i.e. if the input is (t), then y(t) = h(t).By convolution theorem

Y() = G()H()where H() is the system transfer function.

Signal AnalysisSignal Analysis

Signal power• Signal-to-noise ratio (S/N) is an important parameter used

to evaluate the system performance. • Noise, being random in nature, cannot be expressed as a ti

me function, like deterministic waveform. It is represented by power.

Hence, to evaluate the S/N, it is necessary to evolve a method for calculating the signal power.

For a general time domain signal g(t), its average power is given by / 2

2

/ 2

1lim ( )

T

TT

P g t dtT


For a periodic signal, each period contains a replica of the function, and the limiting operation can be omitted as long as T is taken as the period.

For a real signal/ 2

2 2

/ 2

1( ) lim ( )

T

TT

P g t g t dtT

ExampleFind the power of a sinusoidal signal cos0t.Solution / 2

/ 22 0 00 / 2

0/ 2

1 cos 2 sin 21 1 1cos ( ) ( )

2 2 2 2

TT

TT

t tP t dt t

T T

Is it also possible to determine the signal power in frequency domain?

Frequency domain representation for signals of arbitrary waveshapeWhen dealing with deterministic signals, knowledge of the spectrum implies knowledge of the time domain signal.

For an arbitrary (random) signal, Fourier analysis cannot be used because g(t) is not known analytically.

For such an undeterministic signal (which include information signals and noise waveforms), the power spectrum Sg() (or power spectral density) concept is used.


The power spectrum describes the distribution of power versus frequency.

The average signal power is then given by

where Sg() 0 for all .

Another way to evaluate the signal power!

0

1 1( ) ( )

2 g gP S d S d



CorrelationCorrelation measure of similarity between one waveform, andtime delayed version of the other waveform.

The autocorrelation function is a special case of convolution, and it measures the similarity of a function with its delayed replica, and is given by

/ 2*

/ 2

1( ) lim ( ) ( )

T

TT

R g t g t dtT


Important properties of autocorrelation (1) the autocorrelation for = 0 is average power of the signal

The third way to evaluate signal power!

(2) power spectral density Sg() and autocorrelation function of a power signal are Fourier transform pair

/ 2 / 22*

/ 2 / 2

1 1(0) lim ( ) ( ) lim ( )

T T

T TT T

R g t g t dt g t dt PT T

( ) ( )gR S

Questions (Signal Analysis)Questions (Signal Analysis)

1. What type signal is the most fundamental?2. How do you define a periodic signal?3. Does a periodic signal exist within a limited time period?4. Is the message signal a deterministic signal?5. Does negative frequency physically exist?6. What equipments you are going to use in order to observe t

he signal waveform and spectrum?7. How does a pulsed signal differ from a sinusoidal signal?8. What is the frequency domain description of a signal? Is it

more or less useful than the time domain?

Questions (Signal Analysis)Questions (Signal Analysis)

9. Suppose that g1(t) G1() and g2(t) G2(), what is the waveform of g(t) whose Fourier transform is the product of G1() and G2()?

10. How do you measure the similarity between the signal and its delayed replica?

11. How many methods you know for signal power calculation?*12. What is the significance of the time- and frequency- scaling

property of Fourier Transform?

Exercise Problems (Signal Analysis)Exercise Problems (Signal Analysis)

1. Evaluate the integrals

dtte t )(cos

1

2 )( dxxe x

dtte t )3(

dtttt )82)(42( 2

dttt )2()9cos(

2. Simplify the following expressions:(a) [sint/(t + 2)] (t); (b) [1/(j +2)] ( + 3);(c) [sin(k)/] ();

3. Calculate the (a) average value, (b) ac power, and (c) average power of the periodic waveform v(t) = 1 + cos0t.


4. Prove that 1( ) ( )at t

a

5. If g(t) G(), then show that g*(t) G*(-).

6. Find the Fourier transform of the signal f(t) = [A + fm(t)]cosctif fm(t) has a spectrum Fm().

7. If f(t) has a spectrum F(), find the Fourier transform of the following functions: (a) f(t/2 – 5);(b) f(3 – 3t); (c) f(2 + 5t);

8. Determine the average power of the following signals: (a) Acos0t + B sin0t; (b) (A + sin0t) cos0t;


*9. Find the autocorrelation function of the signal, g(t) = Ecos0t;

10. For a power signal, g(t) = Acos(200t)cos(2000t), determine the average power.

Math. TableMath. Table

Trigonometric Identities

2 2 2 2 2 2

2 2

1 1cos sin cos ( ) sin ( )

2 2

sin cos 1 cos 2 2cos 1 1 2sin cos sin sin 2 2sin cos

1 cos 2 1 cos 2cos sin

2 2

sin( ) sin cos cos sin cos( ) cos cos sin sin tan(

j j j j je j e e e ej

tan tan

)1 tan tan

cos( ) cos( ) cos( ) cos( ) sin( ) sin( )cos cos sin sin sin cos

2 2 2

sin sin 2sin cos sin sin 2sin cos2 2 2 2

cos cos 2cos cos cos cos 2sin sin2 2 2

2

Selected Fourier Transform Pairs

0 00 0

0 0 0 0 0 0

( ) 1 1 2 ( )

2 ( ) 2 ( )

cos [ ( ) ( )] sin [ ( ) ( )]

( ) ( )2

j t j t

t

e e

t t j

trect sa

Math. TableMath. Table

Properties of Fourier TransformLinearity: a1g1(t) + a2g2(t) a1G1() + a2G2()Symmetry: If g(t) G(), then G(t) 2g(- )Time scaling:

Time shifting:Frequency shifting:Modulation theorem:Time convolution: g1(t) * g2(t) G1()G2()

Frequency convolution: Conjugate functions: g*(t) G*(-)

Time differentiation:Time integration:

1( ) ( )g at G

a a

00( ) ( ) j tg t t G e

00( ) ( )j tg t e G

0 0 0

1( )cos [ ( ) ( )]

2g t t G G

1 2 1 2

1( ) ( ) ( ) ( )

2g t g t G G

( ) ( )d

g t j Gdt

1

( ) ( ) (0) ( )t

g d G Gj


Classification of signals:Signals can be classified in various ways which are not mutually exclusive:

Continuous (analog) and discrete (digital) signals:Continuous signals are those that do not have any discontinuity in the time domain. Discrete signals are those that assume only specific values at a certain time (and thus have discontinuities).

Information-carrying signals can be either continuous or discrete.

e.g. signals associated with a computer are digital because they take on only two values (binary signals)


Therefore, all signals we have to deal with in

telecommunications are random nonperiodic signals in reality. However,

frequently we will use deterministic periodic signals to demonstrate a point because they

are much easier to work with mathematically.


Communication systems may involve complex waveforms, it is desirable to revolve them in terms of sinusoidal functions. Signal analysis is a tool for achieving this aim.

Principle of signal analysis: To break up all the signals into summations of sinusoidal components. A given signal can be described in terms of sinusoidal frequencies.


The amplitude plot of Fn is a discrete spectrum existing at = 0, 0, 20, 30, … , have amplitudes A/T, (A/T)Sa(/T), (A/T)Sa(2/T), … , etc. respectively.


Interesting phenomena

If the input is a delta function at t = , i.e. it is (t ), then the output is h(t ) and

This means that, convolving a pulse x(t) located near t = 0 with adelta function located at t = has the effect of shifting x(t) to around t = . This also applies in the frequency domain, and is shown schematically below.

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( ) ( ) ( ) ( ) ( )h t h t t t h t

signal and system analysis objectives: to define the energy and power of a signal to classify...

Documents

input signal

output signal

response signal input

time domain

source signal yt

time shift

nfurther properties

input gt