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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 ejωn 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 :
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FAQs on Inverse Discrete Time Fourier Transform - Signals and Systems - Electrical Engineering (EE)
| 1. What is the inverse discrete-time Fourier transform? |  |
| 2. How is the inverse discrete-time Fourier transform different from the Fourier transform? | |
Ans. The Fourier transform converts a time domain signal into its frequency domain representation, while the inverse discrete-time Fourier transform converts a frequency domain representation of a discrete-time signal back into its time domain representation. The Fourier transform is continuous in nature, while the IDTFT deals with discrete-time signals.
| 3. What is the significance of the inverse discrete-time Fourier transform in signal processing? | |
Ans. The inverse discrete-time Fourier transform is widely used in signal processing to analyze and manipulate discrete-time signals in the frequency domain. It allows us to reconstruct the original signal from its frequency components, enabling various applications such as filtering, modulation, and spectral analysis.
| 4. How is the inverse discrete-time Fourier transform computed? | |
Ans. The inverse discrete-time Fourier transform can be computed using the formula:
x[n] = (1/N) * Σ(X[k] * e^(j2πnk/N))
where x[n] is the time domain signal, X[k] is the frequency domain representation, N is the length of the signal, and j is the imaginary unit.
| 5. What are the properties of the inverse discrete-time Fourier transform? | |
Ans. Some important properties of the inverse discrete-time Fourier transform include linearity, time shifting, time reversal, and frequency shifting. Linearity implies that the inverse transform of a sum of signals is equal to the sum of their individual inverse transforms. Time shifting corresponds to shifting the time domain signal without affecting its frequency domain representation. Time reversal flips the time domain signal, while frequency shifting alters the phase of the frequency components.
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