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Signals and Systems

Signals

Signal  A signal is a single valued function of one or more variables that conveys some information.

Signals are represented mathematically as functions of one or more independent variables.

For example – A  speech signal can be represented mathematically by acoustic pressure as a function of time and a picture can be represented by brightness as a function of two spatial variables.

Types of Signals

Signals may be classified in several ways

(a) Continuous Time Signals

A continuous time signal is one which is defined for all values of time.

A continuous time signal needs not be continuous (in the mathematical sense) at all points in time.

e.g., a rectangular wave is discontinuous at several points but it is continuous time signal.

(b) Discrete Time Signals

Discrete time signals are those signals which are defined only at a discrete set of points in time.

e.g., suppose we note down the temperature at a particular place at a fixed time say 5 AM everyday. we get a discrete time signal.

Here, between 1 and 2 we cannot say that value of signal is zero because we recorded temperature only at discrete value. generally, x[nT] is written as x[n] means discrete time signal is denoted by x[n].

(c) Periodic Signals

Periodic signals are those signals which repeat its value at a fixed particular time.

. A continuous time signal x(t) is said to be periodic in time, if

x(t) = x(t + mT)

where, m is any integer.

The smallest value of the positive constant T satisfying the above relation is known as fundamental periode of the periodic signal.

. A discrete time signal x[n] is periodic with period N, where N is a positive integer, if it is unchanged by a time shif of N, i.e., if

x[n] = x[n + N]

for all values of n, above equation holds, then x[n] is periodic with periods N, 2N, 3N,…..Fundamental period NO is the smallest positive value of N.

Example 1. Check whether the given signal is periodic or not.

x(t) = {cos t, if t < 0/sin t, if t > 0

Sol.

  1. sin at and cos at has fundamental period (2/a).
  2. DC term does not affect the fundamental period.

Example 2. is the signal x(t) = 10 cos2 (10 t) a periodic size if it is, what is its fundamental period?

Sol. x(t) = 10 cos2 (10 t)

since, cos 2t = 2 cos2 t  – 1

cos2  t = 1 + cos 2t/2

x(t) = 5 + 5 cos 20 t

since, DC term does not affect fundamental period, so let we take

x(t) = 5 + 5y(t)

y(t) = cos 20 t

fundamental period of this signal is 20/20 {2/a = 1/10

Let z(t) = x(t) + y(t) with x(t) has fundamental period TX and y(t) has fundamental period TY, then fundamental period of z(t) is

T = LCM (TX, TY): if TX/TY is a rational number.

Even Signals

In continuous time, a signal is even if

x(-t) = x(t)

And in discrete time, signal is even if

x[-n] = x[n]

e.g., x(t) = cos t is even signal.

any signal which is symmetris about y-axis is even signal.

Odd Signals

A signal is referred to odd, if

x(-t) = – x(t) and x[-n] = -x[n]

An odd signal must necessarily be 0 at t = 0 or n = 0.

e.g., sin t is an odd signal.

Another Examples

in discrete time, example of even and odd signals are given below.

  1. every signal needs not be even or odd.
  2. any signal which is neither even nor odd can be represented as sum of even and odd signals.

if x(t) is a signal which is neither even nor odd. then x(t) can be written as

X(t) = XE (t) + XO(t)

XE(t) = x(t) + x(-t)/2

xo(t) = x(t) – x (-t)/2

Example 3. Find the even and odd components of x(t) = ejt

Sol. let, xe(t) and xo(t) be the even and odd components of ejt respectively.

ejt = xe (t) + xo(t)

xe(t) = 1/2 [ejt + e-jt] = cos t

x0(t) = 1/2 [ejt – e-jt] = j sin t

Energy Signals and Power Signals

for an arbitrary continuous time signal x(t), the normalized energy content E of x(t) is defined as

E = |X(t)2 dt

and normalized average power P of x(t) is defined as

P = lim 1/T |x(t)2| dt

similarly, for a discrete-time signal x[n], the normalized energy content E of x[n] is defined as

E = |x[n]|2

the normalized average power P of x[n] is defined as

P = lim 1/2N + 1n = – N |x[n]|2

  1. x(t) (or x[n]) is said to be an energy signal (or sequence) if and only if 0 < E OO and P = 0
  2. x(t) (or x[n]) is said to be a power signal (or sequence ) if and only if 0 < p < oo, thus implying that E = OO
  3. Signals that satisfy neither property are preferred to as neither energy signals nor power signals.

Deterministic and Random Signals

Deterministic signals are those signals whose values ar completely specified for any given time. thus, a deterministic signal can be modelled by a known function of time t.

e.g.,        x(t) = 10 cos 100 t

At any instant of time one can calculate the value of x(t) from the given expression of x(t).

Random signals are those signals that take random values at any given time and must characterized statistically.

e.g., noise signals and EEG signals.

Random signals, if stationary may be described in terms of certain average values only.

One-dimensional and Multi-dimensional Signals

A signal which is a function of only one variable is referred to as a one-dimensional signal.

x(t) = a cos 0o t.

A signal which is a function of more than one variables is known as multi-dimensional signal.

e.g., image of any object, intensity of light reflected from any point.

Representation of Signals

(a) Continuous Time Signals

Continuous time signals are specified for all values of time by a function of time like, x(t) = e-|t|; –oo < t < oo or by a look up table which specifies its values for all time range. for example,

x(t) = {2, – oo < t < 0/1, 0 < t < oo

they are represented diagrammatically by their waveforms, i.e., graphs depicting their varition with time.

(b) Discrete Time Signals

A discrete time signals is a sequence of numbers. these number are assumed to occurring at regular intervals of T second. the nth sample of a discrete time signal is generally denoted by x[nT] or simply x[n].

A discrete time signal x[n] may be specified or described in one of the following ways

(i) by means of a look-up table

(ii) by means of an equation which specifies the nth and sample values x[n] as a function of n.

for example, x[n] = n2oo < n < oo

(iii) by means of a recursive formula such as

x[n] = 2x [n – 1] + [n] with x(-1) = 0

where, [n] = {z, for n = 0/0, otherwise

Some Commonly Used Signals for Continuous Time Signals

  1. The unit impulse function (t)

Unit impulse function is defined as

(t) = {0  t = 0/ 1 t = 0

(t) dt = 1

Properties of unit impulse function

x(t) (t) dt = x(0)

the area under unit impulse function is equal to 1.

(t) dt = 1

the width of (t) along the time axis is zero.

Sampling property

if x(t) is continuous at t

x(t) (t) dt = x(t) t  x(t)

for any t1 and t2 such that the interval t1 to t2 includes t

x(t) dt

= x(t) dt = x

from eqs. (1) and (2),

x(t) (t-) dt

this is called sampling property of the impulse function.

x(t) (t – to) = x(to) (t – to)

same as sampling property.

  1. Unit step function u(t)

the unit step function is defined by

u(t) = {1, for t > 0/0, for t < 0

u(t – to) = {1, for t < to/0, for t < to

Relationship between unit impulse and unit step functions

u(t) = d

(t) = du(t)/dt

  1. Unit ramp function

Unit ramp function is defined by

r(t) = {t, for t > 0/0, otherwise

Relationship between unit ramp and unit step functions

r(t)  u(t) dt and u(t) = d/dt r(t)

  1. The exponential signal

the continuous time exponential signal is defined by

x(t) = Ceat

where, c and a are complex. numbers.

if we take c and a as real, then it becomes real exponential signal.

x(t) = CRet

CR = Real part of C

a = real part of a

C = CR + jCI

a = 0 + joo

  1. the rectangular pulse

A rectangular pulse, symmetrically located with respect to the time origin having an amplitude A and duration T is described by

x(t) = {A, – T/2 < t < T/2 0, otherwise

it is given a special symbol

area (t/T) or A (t/T)

A represents amplitude.

t represents function of time.

  1. Sinusoidal signal

A continuous time sinusoidal signal can be expressed as

x(t) = A sin (0o t + 0)

A = amplitude

00 = angular frequency

0 = phase angle in radian

it is periodic function with fundamental period 2/00 = To

Discrete Time Signals

  1. The unit sample sequence

it is denoted by [n] and is defined by

[n] = {1, for n = 0/0, otherwise

  1. The unit step sequence

the unit step sequence is defined by

u[n] = (1, for n > 0/0, for n > 0

  1. The unit ramp sequence

A unit ramp sequence is denoted by and is defined by

r[n] = [n, n > 0/0, n < 0

Relation between unit step and unit impulse sequences

[n] = u[n] = – u[ n – 1]

u[n] = [k]

x[n] = x[k] [n – k]

Properties of unit sample sequence

x[n] [n] = x[0] [n]

x[n] [n – n0] = x[n0] [n – no]

  1. The exponential sequence

The exponential sequence is defined by x[n] = an u[n] where, a is constant.

  1. Cosinusoidal sequence

A discrete time cosinusoidal function is defined by

x[n] = A cos n

Operations on Signals (Including Transformation of Independent Variables)

(a) Continuous Time Signals

  1. Addition and subtraction

let x(t) and y(t) be two signals, then their sum z(t) is defined by

z(t) = x(t) + y(t)

  1. Multiplication of signal by a constant

z(t) = a x(t)

if |a|>1; then it is known as amplitude scaling.

if|a| <1; then it is n attenuated system.

  1. Multiplication of two signals

the  operation is defined by

z(t) = y(t) x(t)

it is very useful in communication system.

  1. Differentiation and integration

for example , the voltage V(t) across an inductance L when a current I(t) flows through it, is given by

V(t) = L dV(t)/dt

voltage across capacitor is given by

V(t) = 1/C

  1. Shifting in time

Consider a signal x(t). then, the signal x(t – to) represents a delayed version of x(t) delayed by 2 s, is shown as

  1. Compressing and expanding a signal in time

let there is a signal x(t).

then x(at) shows expanding of a < 1 and x(at) shows compressing of a > 1.

in case of time shifting and scaling both then, first shift the signal and then scaling is done.

Example 4. plot x(2t – 3) if x(t) is given below,

Sol. first do time shift.

if time shifting scaling and time reversal have to be done, then

perform time shifting first

then time scaling

then time reversal

Example 5. x(t) is shown below, plot x(-2t + 1).

(b) Discrete Time Signals

  1. Time scaling

Consider x[n] as shown below, then find x[2n].

  1. Time Shifting

same as in CT signals.

  1. Addition

z[n] = x[n] + y[n]. same as in CT.

  1. Multiplying the signal by a constant

when a discrete time signal x[n] is to be multiplied by a constant, say a, we multiply every sample of x[n] by that constant.

Intro Exercise – 1

  1. The period of signal x(t) = 24 + 50 cos 60t is

(a) 1/30s

(b) 60s

(c) 1/60s

(d) not periodic

  1. The period of signal x(t) = 10 sin 5t – 4 cos 9t is

(a) 24/35

(b) 4/35

(c) 2

(d) not periodic

  1. The period of signal x(t) = 5t – 2 cos 6000 t is

(a) 0.96 ms

(b) 1.4 ms

(c) 0.4 ms

(d) not periodic

  1. The period of signal x(t) = 4 sin 6t + 3sin 3t is

(a) 2/3s

(b) 2/3s

(c) 2

(d) not periodic

  1. Consider the following signals

x(t) = cos t + 2 cos 3 t + 3 cos 5 t

y(t) = sin 2t + 6 cos 2 t

z(t) = sin 3t cos 4t

periodic signals are

(a) x(t) and y(t)

(b) y(t) and z(t)

(c) x(t) and z(t)

(d) All of these

  1. The signal x(t) = e-4t u(t) is a

(a) power signal with poo = 1/4

(b) power signal with Poo = 0

(c) energy signal with Eoo = 1/4

(d) energy signal with Eoo = 0

  1. The signal x(t) = ej(2t + 6) is a

(a) power signal with Poo = 1

(b) power signal with Poo = 2

(c) energy signal with Eoo = 2

(d) energy signal with Eoo = 1

  1. The raised cosine pulse x(t) is defined as x(t) = {1/2 cos ot + 1), – <t < otherwise

the total energy of x(t) is

(a) 3/4

(b) 3/8

(c) 3

(d) 3/2

  1. Consider the voltage waveform shown below:

the equation for V(t) is

(a) u(t – 1) + u(t – 2) + u(t – 3)

(b) u(t – 1) + 2u(t – 2) + 3u(t – 3)

(c) u(t – 1) + u(t – 2) + u(t – 4)

(d) u(t – 1) + u(t – 2) + u(t – 3) -3u(t – 4)

  1. Consider the following function for the rectangular voltage pulse shown below
  2. V(t) = u(a – t) . u(t – b)
  3. V(t) = u(b – t). u(t – a)
  4. V(t) = u(t – a) -u(t -b)

the functions that describe the pulse are

(a) 1 and 2

(b) 2 and 3

(c) 1 and 3

(d) all of these

  1. A signal is described by x(t) = r(t – 4) -r(t – 6), where r(t) is a unit ramp function starting at t = 0. the signal x(t) is represented as
  2. The trapezoidal pulse y(t) is related to the x(t) as y(t) = x(10t – 5)

the sketch of y(t) is

  1. The trapezoidal pulse x(t) is time scaled producting y(t) = x(5t). the sketch for y(t) is
  2. The trapezoidal pulse x(t) is time scaled producting y(t) = x(t/5). the sketch for y(t) is
  3. The trapezoidl pulse x(t) is applied to a differentiator, defined by y(t) = dx(t)/dt. the total energy of y(t) is

(a) 0

(b) 1

(c) 2

(d) 3

  1. The total energy of x(t) is

(a) 0

(b) 13

(c) 13/3

(d) 26/3

  1. u[n] + u[-n] is equal to

(a) 2

(b) 1 + [n]

(c) 2 + [n]

(d) 1

  1. A discrete time signal is given as below

x[n] = cos 9 + sin (n/7 + 1/2)

the period of signal is

(a) periodic with period N = 126

(b) periodic with period N = 32

(c) periodic with period N = 252

(d) not periodic

  1. A discrete time signal is given as below

x[n] = cos (n/8) cos (n/8)

(a) periodic with period 16

(b) periodic with period 16 + 1

(c) periodic with period 8

(d) not periodic

  1. A discrete time signal is given as below

x[n] = cos (n/2) – sin (n/8) + 3cos (n/4 + 3)

the period of signal is

(a) periodic with period 16

(b) periodic with period 4

(c) periodic with period 2

(d) not periodic

Answers with Solutions

  1. (a)

2/t = 60         T = 1/30

  1. (c)

2/t1 = 5           T1 = 2/5 and 2/T2 = 9

T2 = 2/9

LCM (2/5, 2/9) = 2

  1. (d)

not periodic because of t.

  1. (d)

not periodic because least common multiple is infinite.

  1. (c)

y(t) is not periodic although sin t and 6 cos 2t are independently periodic. the fundamental frequency cannot be determined.

  1. (c)

this is energy signal because

Eoo = |x(t)|2 dt < oo = e-4t u(t) dt = [e-4t dt = 1/4

  1. (a)

|x(t)| = 1, EOO = |x(t)|2 dt = oo

so, this is a power signal not a energy.

P00 = lim 1/2t  x(t)2 dt = 1

  1. (a)

E = 1/4 cos (ot + 1)2 dt

= 1/2 (1/2 cos 2ot + 1/2 + c cos ot + 1) dt

= 1/2 (3/2) = 3/4

  1. (d)

V(t) is sum of three unit step signals starting from 1, 2, and 3, all signals-end at 4.

  1. (b)

The function 1 does not describe the given pulse. it can be shown as follows

  1. (b)

the figure is as shown below.

  1. (c)

the figure is as shown below

  1. (d)

multiplication by 5 will bring contraction on time scale. it may be checked by x(5 x 0.8) = x(4).

  1. (a)

division by 5 will bring expansion on time scale. it may be checked by

y(t) = x(20/5) = x(4)

  1. (c)

y(t) = {1,  for -5 < t <-4/-1,  for 4 < t < 5/0, otherwise

E = (1)2 dt + (-1)2 dt = 2

  1. (d)

E = 2  x2 (t) dt

= 2 (1)1 dt + 2  (5 – t)2 dt = 8 + 2/3 = 26/3

  1. (b)

the pole are as follows :

  1. (a)

both signals are periodic N1 = 18 N2 = 14

N = LCM (18, 14) = 126

  1. (d)

cos (n/8) is not periodic. so, x[n] is not periodic.

  1. (a)

N1 = 4, N2 = 16, N3 = 8, N = LCM (4, 16, 8) = 16