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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