Basic signals and systems Signals and systems

signal is a set of data or information Eg. Telephone voice, monthly sales of corporation

signals are a functions of the independent variable time and space.

Signals are processed by systems

System is an entity that processes a set of information(generally inputs) and yields another signals (generally outputs)

Systems may be made up of physical components (electrical, mechanical, e.t.c)

Classification of signals ex00basicsignalclassification.m

continuous time and discrete time signal - - based on the nature of signal along the time axis . - signal is defined for a range in time or instants of time Eg: telephone signal vs. monthly sales of company

Analog and digital signals

- based on the nature of signal along the amplitude - Amplitude is taking infinite possible values or finite possible

values

Periodic and aperiodic signals

Periodicity condition is x(t) = x(t+m) for all t , where m is the the smallest value that satisfies the

periodicity condition and it is called the fundamental period.

Properties - Periodic signal must start at t = - and continue for ever - Periodic signal x(t) can be generated by periodic extension of

any segment x(t) of duration m (the fundamental period) - Area under x(t) [periodic signal] over any interval of duration

m is the same. i.e for any real numbers a and b

+

() =

+

()

Causal and non causal signals

- A signal that does not start before t=0 is a causal signal i.e. x(t) = 0 for t < 0

- A signal that exists before t=0 is a non-causal signal i.e. x(t) 0 for t < 0

- Everlasting signal is always non-causal, but non-causal signal is not necessarily everlasting

Energy and Power signals

o A signal with finite energy is the energy signal o A signal with finite power is the power signal

Power is the time average of energy A signal cannot be both an energy signal and power signal A ramp signal is neither energy nor power signal

Deterministic and random signals

- Deterministic signals :A physical description is known completely either in mathematical form or a graphical form. The nature and amplitude of such signal at any time can be predicted.

o Examples are x(t) = bt and x(t) = a sint - random signals: amplitude cannot be predicted precisely but

known only in terms of probabilistic description. o A typical example of random signals is thermal noise

generated in electric circuit.

Even and Odd parts of the signal:

A real function xe(t) is said to be an even function of t if

xe(t) = xe(-t) i.e symmetrical about vertical axis at t= 0

A real function xo(t) is said to be an odd function of t if

xo(t) = -xo(-t) i.e anti -symmetrical about vertical axis at t= 0

o Some properties:

Even function odd function = odd function

Odd function odd function = Even function

Even function Even function = Even function

Area

() = 2

0() and

0

() =0

Given any arbitrary signal x(t), odd and even parts can be found as

Even part of the signal xe(t) = 1

2[x(t)+x(-t)]

Odd part of the signal xo(t) = 1

2[x(t)-x(-t)]

MATLAB code: ex01signalevenodd.m

What is the signal size of these signals ?

Size of a signal: -----------> It indicates the largeness or strength of the

signal

Eg: human size -----> volume ; not the height only

Signal energy: Ex = 2

() OR Ex = |()|2

o Signal size is the area under square of the signal x(t), i.e. x2(t) o Signal energy should be finite and non zero for it to be a

meaningful measure of signal size o Necessary condition is that amplitude of signal x(t) ----> 0 as

|| ---> Examples ??? x(t) = 2 : -1 t 0 2 e-t/2 : t 0

Energy of the signal is 8.0018 units

Signal power: o If the amplitude of the signal x(t) is not ----> 0 as || ---> , signal

energy is infinite. o A more meaningful measure of signal size in such a case would be

Time average of the energy

o Px = lim

1

2()

2

2

OR Px = lim

1

|()|2

2

2

where T is the period of the periodic signal. o Signal power should be finite for it to be a meaningful measure of

signal size.

Power of the signal is 0.3333 units

= RMS Value of the signal, generally applicable for periodic signal.

Note: Generally mean of an entity averaged over a large time interval approaching infinity, exists if the entity either is periodic or has a statistical regularity. If such a condition is not satisfied, the average may not exists.

Power is the time average of energy A signal cannot be both an energy signal and power signal A ramp signal is neither energy nor power signal

A causal signal but periodic is also referred as power signal.

Matlabcode : ex01signalenergypower.m

Example : determine the power and RMS value of X(t) = C cos (ot +) and X(t) = C1 cos (ot +1) + C2 cos (ot +2) with 1 2

........................................

Px = 12

2 +22

2 OR Px = 1

2

2=1

signal operation:

Amplitude operation : 2 x(t) OR 2 + x(t) e.t.c.

Time operation o Time shifting :

delay -----> right shift Advance ------> left shift

o Time scaling : Compression -----> fast up ,

what happens to x(t) at some instant t also happens to (t) = x(at) at the instant t/a , where a > 0

Expansion -----> slow down ,

what happens to x(t) at some instant t also happens to (t) = x(t/a) at the instant at, where a > 0

o Time reversal :

What happens to x(t) at some instant t also happens to (t) = x(-t) at the instant t.

o Arbitrary operation : x(-2t-3)

= x(-2(t-[3/2])) ---------------------

Arrange the each operations in multiplication form and then the sequence of operations are Reversal Compress by 2 Advance by 3/2

MATLAB code: ex01signal1operation.m

Basic signals models - CONTINUOUS TIME

Unit step functions u(t)

o u(t) = {1 00 < 0

delayed u(t-) = {1 0 <

o If we have any arbitrary everlasting signal to start at t = 0, we need to multiply the signal by u(t), to get the causal signal. Eg. e-at is an everlasting signal but e-atu(t) is a causal signal.

o A pulse with nonzero value from 1s to 2s can be expressed in terms of step functions as u(t-1) u(t-2)

OR u(t-1) *u(2-t)

o It is useful in specifying a function with different mathematical description over different intervals.

x(t) = t for 0 t 1 and e-t for 1 t 4 can be written in single mathematical expression as x(t) = t { u(t) u(t-1) } + e-t {u(t-1) u(t-4) }

Example :

MATLAB code: ex00basicsignalmodels1.m

Unit Impulse functions (t) :

It is defined as

i)

() = 1 i.e area under the impulse curve is one

ii) (t) = 0 for t 0 i.e. as t -----> 0, the value and shape of the impulse curve is not defined.

Example

Impulse function does not define an unique function

Delayed impulse is (t-) , i.e delay by units Multiplication of a function by an impulse :

Let x(t) be an arbitrary continuous time signal. Then, x(t) (t) = x(0) (t) is an impulse of strength x(0) at t= 0 x(t) (t-T) = x(T) (t-T) is an impulse of strength x(T) at t= T

Sampling property of the unit impulse function

()

() = x(0)

() since (t) = 0 for t 0

= x(0) since

() = 1

Impulse function can be defined in terms of its effect on a test function x(t).

It is not a true function in ordinary sense

Its range is undefined

In the generalised sense ( considering the unit step function),

= (t) OR

()= u(t)

u(t) is discontinuous and hence

does not exist in ordinary

sense.

The exponential functions

o x(t) = et where is complex in general given by = +j

o et = e ( +j)t = e t ejt = e t (cos t+jsin t )

o et is the generalisation of the function ejt where the frequency

variable j is generalised as complex frequency variable .

o Function, et compasses Large class of functions : viz

1. = 0 i.e. A constant K = K e0t

2. = i.e. with = 0 A monotonic exponential e t

is +ve or ve real values

3. = j i.e. with = 0 A Sinusoid cos t

4. = +j A exponentially varying sinusoid

e t (cos t )

Matlab : ex00basicsignalmodels2.m

Other commonly used standard signals :

A unit rectangular signal

A unit triangular signal

Sinc function

Signum function

Half triangle

A unit rectangular signal rect (x) =

{

0 || >

1

21

2|| =

1

2

1 || physical or non anticipative

System output at any instant to depends only on the value of the input x(t) for t < to i.e conversely, present o/p of the system does not depend on future value of input.

o Non causal ----> anticipative System output at any instant to depends on the future value of the input x(t) i.e for t > to. i.e conversely, the response starts before the input is applied to the system.

o Prophetic system. o Generally non temporal systems. i.e. system does not

depends on time. Eg. optics, charge etc. o Noncausal systems are not realisable systems.

Example: y(t) = 10 x(2t) is a Non causal system Put t = -2 y(-2) = 10 x(-4) ----- input before the o/p

Put t = 0 y(0) = 10 x(0) ----- o/p at same instant as i/p Put t = 2 y(2) = 10 x(4) ----- o/p before the input which gives Non-causal property

Continuous time and discrete time systems System whose inputs and outputs are continuous time signals are continuous time system. System whose inputs and outputs are discrete time signals are discrete time system.

Analog and digital systems System whose inputs and outputs are analog in nature of amplitude are analog systems. System whose inputs and outputs are digital in nature of amplitude are digital systems.

Invertible and non-invertible systems o A system S performs certain operation(s) on input signal(s). If

we can obtain the same input(s) back from the corresponding output, by some operation, the system S is said to be invertible

o For an invertible system, it is essential that every input have a unique output.

o System that achieves the inverse operation is the inverse system for S.

o If S is ideal integrator, then ideal differentiator is the inverse system.

o Also,

For lossless coding, the input to the encoder must be exactly

recoverable from the output. It means that the encoder must be invertible.

o It is something related to the concept of identity system o Otherwise it is Non-invertible systems.

Eg. Rectifiers where y(t) = |x(t)|

Stable and unstable systems -can be internal / external o If the signal x(t) is bounded, then its magnitude is always a finite

value. Mathematically |x(t)| Mx < , where M is a positive real finite number.

Ex: sinusoidal signal , OR Exponential decay signal o Any signal which does not satisfy |x(t)| Mx < is called

unbounded signal. -------- BIBO stability (External) o For this to happen, output of the system is also bounded i.e.

|y(t)| My < for all values of t. o Other class of stability is the internal which mainly refers

whether the behaviour is stable.

Single input single output system (SISO), MIMO

System models: (input-output description) It is basically a mathematical expression OR rule that satisfies

and approximates the dynamic behaviour of the system. ORDER of a SYSTEM : the order of a continuous time system corresponds to the total number of dynamic elements or highest derivative of the output signal which may appear in the input-output differential equation. Note: Most basic characterisation of system is a linear, time invariant (LTI) system . The reason for this are

1. Powerful analysis techniques are exists for such systems only.

2. Many real world systems can be closely approximated as LTI systems.

3. Analysis techniques for LTI systems can be generalised to any extent and it suggest approaches for the analysis of no-linear systems.

Differential equation ()

+ 1

1()

1+ . . +1

()

+ () =

0()

+ 1

1()

1+ . .+1

()

+ ()

OR Q(D) y(t) = P(D) x(t)

Where, P(D) and Q(D) are respective operator polynomials for input

and output.

Transfer function model

()

()= b0D

m+b1Dsm1+ b2D

m2++bm1D+bm

Dn+a1Dn1+ a2Dn2++an1D+an

() = ()

()=()

()

Frequency response model

() =()

()

State space model

= (, , ) ----------> State equation = (, , ) -----------> Output equation

Where, x State variables ; u Input signal ;

f and g are functions

Basic signal plotting and signal operation

MATLAB code:

Plotting using inline functions : { >> prompt is used in every following MATLAB Statements} Many simple functions are most conveniently represented by using

MATLAB

inline objects.

Ex: Consider a continuous time function: f(t) = e-tcos(2t) can be expressed as, >>f= inline('exp(-t).*cos(2*pi*t)','t')

Once defined, f(t) can be evaluated simply by passing the input values

of interest.

Ex: >> t= 0; >> f(t) ans = 1

f(t) can be plotted over the interval -2 t 2 as >> t= [-2:.01:2];

>> f(t) ans = 100 values of f(t) .

>> plot(t,f(t)); will plot the above function.

>> u=inline('t>=0','t') will create a unit step function.

>> p=inline('(t>=0)&(t=-2)&(t=0)&(t=1)&(t

Even part of the signal xe(t) =

[x(t)+x(-t)]

Odd part of the signal xo(t) =

[x(t)-x(-t)]

MATLAB code: ex01signalenergypower.m