Unit – ISignals and systems

Continuous time signalsDiscrete time signalsTransformations of the independent variableExponential and sinusoidal signalsThe unit impulse and unit step functionsContinuous time and discrete time systemsBasic system propertiesLinear time invariant systemsThe discrete time LTI systems – The convolution SumThe continuous time LTI systems – The convolution IntegralProperties of linear time invariant systemsCausal LTI systems described by differential and difference equationsSingularity functions

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

1. Speech signal

(acoustic pressure as function of time)

2. Picture signal

(brightness as a function of two variables)

3. In geophysics

(Density, porosity and electrical resistivity)

4. In Metrology

(air pressure, temperature, wind speed with

altitude)

Continuous time signalsIn continuous time signal the independent variable is continuous and thus these signals are defined for a continuum of values of independent variable

Example: 1. Speech as a function of time2. Atmospheric pressure as a

function of altitude

Discrete time signals

Discrete time signals are only defined at discrete times and for these signals the independent variable takes only discrete set of values.Example:1.Weekly stock market index2. Literacy versus population

Transformation of independent variable

Time reversalIf x(t) represent an audio signal in tape recorder then x(-t) is the same audio signal played from backward.

Time ScalingIf x(t) is a audio signal then x(2t) represent the same audio played at twice the speed and x(t/2) represent that audio is played at half the speed.

Time shiftingIf x(t) is a signal then x(t-2) is the right shift of x(t) by 2 and x(t+2) is the left shift of x(t) by 2

Even and odd signalsA signal is said to be even if it is identical with its reflection about the origin.X(-t) = x(t)X[-n] = x[n]

A signal is said to odd if its is not identical with its reflection about the originX(-t) = -x(t)X[-n] = x[n]

The even and odd part of a signal can be found by

Even{x(t)} = ½ {x(t) + x(-t)}

Odd{x(t)} = ½ {x(t) – x(-t)}

Periodic and non periodic signalA signal is said to periodic if it satisfies the conditions

X(t) = X( t + T), for all tX[n] = X[ n + N], for all n

The smallest value of T and N for which the above equation exists is called fundamental frequency.

Continuous time complex exponential and

sinusoidalX(t) = C eat if C and a are real, then x(t) is called real exponential.

If a>0, then x(t) is growing exponentialIf a<0, then x(t) is decaying exponentialIf a=0, then x(t) is a constant signal

Purely imaginary exponential

X(t) = ejw0

t

If w0 = 0, then x(t) = 1. that is x(t) is period for any value of T

If w0 is not equal to zero, then x(t) has fundamental period

Sinusoidal signalX(t) = A Cos (w0t + φ)