Inverse Discrete Time Fourier Transform

Inverse DTFT : 

DTFT of a discrete periodic signal x[n] by period N is given by : 

:

The Fourier coefficients of this periodic function are given by

The above equation is referred as Inverse DTFT equation .

Inverse DTFT equation : 

Now ,
x[-n] is the nth Fourier series co-efficient of 

Now,

can be interpreted  as X(ω) multifplied by  the vector of e^jωn and summed over 

Now inverse DTFT for the cross co-relation between sequences x[n] and h[n] can be written as :

i.e dot product of sequences x[n] and h[n] = dot product of DTFT's of x[n] and h[n] . in particular put x[n] =h[n] , then

The above is Parseval's relation for discrete time periodic signals .

here  isthe total energy in x['] and   is the energy  spectral density of x['].

Conclusion:

In this lecture you have learnt:

• DTFT is periodic in  ω with period  2π  and also , the I-DTFT is periodic in ω with period 2π   .
•  THE DTFT                   
• Parseval's Relation for Discrete periodic signals :