- Digital Signal Processing Tutorial
- DSP - Home
- DSP - Signals-Definition
- DSP - Basic CT Signals
- DSP - Basic DT Signals
- DSP - Classification of CT Signals
- DSP - Classification of DT Signals
- DSP - Miscellaneous Signals
- Operations on Signals
- Operations Signals - Shifting
- Operations Signals - Scaling
- Operations Signals - Reversal
- Operations Signals - Differentiation
- Operations Signals - Integration
- Operations Signals - Convolution
- Basic System Properties
- DSP - Static Systems
- DSP - Dynamic Systems
- DSP - Causal Systems
- DSP - Non-Causal Systems
- DSP - Anti-Causal Systems
- DSP - Linear Systems
- DSP - Non-Linear Systems
- DSP - Time-Invariant Systems
- DSP - Time-Variant Systems
- DSP - Stable Systems
- DSP - Unstable Systems
- DSP - Solved Examples
- Z-Transform
- Z-Transform - Introduction
- Z-Transform - Properties
- Z-Transform - Existence
- Z-Transform - Inverse
- Z-Transform - Solved Examples
- Discrete Fourier Transform
- DFT - Introduction
- DFT - Time Frequency Transform
- DTF - Circular Convolution
- DFT - Linear Filtering
- DFT - Sectional Convolution
- DFT - Discrete Cosine Transform
- DFT - Solved Examples
- Fast Fourier Transform
- DSP - Fast Fourier Transform
- DSP - In-Place Computation
- DSP - Computer Aided Design
- Digital Signal Processing Resources
- DSP - Quick Guide
- DSP - Useful Resources
- DSP - Discussion
DSP - DFT Solved Examples
Example 1
Verify Parseval’s theorem of the sequence $x(n) = \frac{1^n}{4}u(n)$
Solution − $\displaystyle\sum\limits_{-\infty}^\infty|x_1(n)|^2 = \frac{1}{2\pi}\int_{-\pi}^{\pi}|X_1(e^{j\omega})|^2d\omega$
L.H.S $\displaystyle\sum\limits_{-\infty}^\infty|x_1(n)|^2$
$= \displaystyle\sum\limits_{-\infty}^{\infty}x(n)x^*(n)$
$= \displaystyle\sum\limits_{-\infty}^\infty(\frac{1}{4})^{2n}u(n) = \frac{1}{1-\frac{1}{16}} = \frac{16}{15}$
R.H.S. $X(e^{j\omega}) = \frac{1}{1-\frac{1}{4}e-j\omega} = \frac{1}{1-0.25\cos \omega+j0.25\sin \omega}$
$\Longleftrightarrow X^*(e^{j\omega}) = \frac{1}{1-0.25\cos \omega-j0.25\sin \omega}$
Calculating, $X(e^{j\omega}).X^*(e^{j\omega})$
$= \frac{1}{(1-0.25\cos \omega)^2+(0.25\sin \omega)^2} = \frac{1}{1.0625-0.5\cos \omega}$
$\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{1}{1.0625-0.5\cos \omega}d\omega$
$\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{1}{1.0625-0.5\cos \omega}d\omega = 16/15$
We can see that, LHS = RHS.(Hence Proved)
Example 2
Compute the N-point DFT of $x(n) = 3\delta (n)$
Solution − We know that,
$X(K) = \displaystyle\sum\limits_{n = 0}^{N-1}x(n)e^{\frac{j2\Pi kn}{N}}$
$= \displaystyle\sum\limits_{n = 0}^{N-1}3\delta(n)e^{\frac{j2\Pi kn}{N}}$
$ = 3\delta (0)\times e^0 = 1$
So,$x(k) = 3,0\leq k\leq N-1$… Ans.
Example 3
Compute the N-point DFT of $x(n) = 7(n-n_0)$
Solution − We know that,
$X(K) = \displaystyle\sum\limits_{n = 0}^{N-1}x(n)e^{\frac{j2\Pi kn}{N}}$
Substituting the value of x(n),
$\displaystyle\sum\limits_{n = 0}^{N-1}7\delta (n-n_0)e^{-\frac{j2\Pi kn}{N}}$
$= e^{-kj14\Pi kn_0/N}$… Ans
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