Illustration of the Discrete Fourier Tranform DFT

Mathematica 8:

Do[
nn = i;
Print[MatrixForm[Transpose[Table[{1}, {n, 1, nn}]]]]
Print[MatrixForm[
Table[Table[Cos[-2*Pi*(n – 1)*(k – 1)/nn], {k, 1, nn}], {n, 1,
nn}]]]
Print[MatrixForm[
Chop[N[Table[
Total[Table[Cos[-2*Pi*(n – 1)*(k – 1)/nn], {k, 1, nn}]], {n, 1,
nn}]]]]]
, {i, 1, 12}]

Signal or time domain

\left(  \begin{array}{c}   1  \end{array}  \right)

Dicrete Fourier Cosine Transform

\left(  \begin{array}{c}   1  \end{array}  \right)

Spectrum or frequency domain

\left(  \begin{array}{c}   1.  \end{array}  \right)

Signal or time domain

\left(  \begin{array}{cc}   1 & 1  \end{array}  \right)

Dicrete Fourier Cosine Transform

\left(  \begin{array}{cc}   1 & 1 \\   1 & -1  \end{array}  \right)

Spectrum or frequency domain

\left(  \begin{array}{c}   2. \\   0.  \end{array}  \right)

Signal or time domain

\left(  \begin{array}{ccc}   1 & 1 & 1  \end{array}  \right)

Dicrete Fourier Cosine Transform

\left(  \begin{array}{ccc}   1 & 1 & 1 \\   1 & -\frac{1}{2} & -\frac{1}{2} \\   1 & -\frac{1}{2} & -\frac{1}{2}  \end{array}  \right)

Spectrum or frequency domain

\left(  \begin{array}{c}   3. \\   0. \\   0.  \end{array}  \right)

Signal or time domain

\left(  \begin{array}{cccc}   1 & 1 & 1 & 1  \end{array}  \right)

Dicrete Fourier Cosine Transform

\left(  \begin{array}{cccc}   1 & 1 & 1 & 1 \\   1 & 0 & -1 & 0 \\   1 & -1 & 1 & -1 \\   1 & 0 & -1 & 0  \end{array}  \right)

Spectrum or frequency domain

\left(  \begin{array}{c}   4. \\   0. \\   0. \\   0.  \end{array}  \right)

Signal or time domain

\left(  \begin{array}{ccccc}   1 & 1 & 1 & 1 & 1  \end{array}  \right)

Dicrete Fourier Cosine Transform

\left(  \begin{array}{ccccc}   1 & 1 & 1 & 1 & 1 \\   1 & \frac{1}{4} \left(-1+\sqrt{5}\right) & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{4} \left(-1+\sqrt{5}\right) \\   1 & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{4} \left(-1+\sqrt{5}\right) & \frac{1}{4} \left(-1+\sqrt{5}\right) & \frac{1}{4} \left(-1-\sqrt{5}\right) \\   1 & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{4} \left(-1+\sqrt{5}\right) & \frac{1}{4} \left(-1+\sqrt{5}\right) & \frac{1}{4} \left(-1-\sqrt{5}\right) \\   1 & \frac{1}{4} \left(-1+\sqrt{5}\right) & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{4} \left(-1+\sqrt{5}\right)  \end{array}  \right)

Spectrum or frequency domain

\left(  \begin{array}{c}   5. \\   0. \\   0. \\   0. \\   0.  \end{array}  \right)

Signal or time domain

\left(  \begin{array}{cccccc}   1 & 1 & 1 & 1 & 1 & 1  \end{array}  \right)

Dicrete Fourier Cosine Transform

\left(  \begin{array}{cccccc}   1 & 1 & 1 & 1 & 1 & 1 \\   1 & \frac{1}{2} & -\frac{1}{2} & -1 & -\frac{1}{2} & \frac{1}{2} \\   1 & -\frac{1}{2} & -\frac{1}{2} & 1 & -\frac{1}{2} & -\frac{1}{2} \\   1 & -1 & 1 & -1 & 1 & -1 \\   1 & -\frac{1}{2} & -\frac{1}{2} & 1 & -\frac{1}{2} & -\frac{1}{2} \\   1 & \frac{1}{2} & -\frac{1}{2} & -1 & -\frac{1}{2} & \frac{1}{2}  \end{array}  \right)

Spectrum or frequency domain

\left(  \begin{array}{c}   6. \\   0. \\   0. \\   0. \\   0. \\   0.  \end{array}  \right)

Signal or time domain

\left(  \begin{array}{ccccccc}   1 & 1 & 1 & 1 & 1 & 1 & 1  \end{array}  \right)

Dicrete Fourier Cosine Transform

\left(  \begin{array}{ccccccc}   1 & 1 & 1 & 1 & 1 & 1 & 1 \\   1 & \text{Sin}\left[\frac{3 \pi }{14}\right] & -\text{Sin}\left[\frac{\pi }{14}\right] & -\text{Cos}\left[\frac{\pi }{7}\right] & -\text{Cos}\left[\frac{\pi }{7}\right] & -\text{Sin}\left[\frac{\pi }{14}\right] & \text{Sin}\left[\frac{3 \pi }{14}\right] \\   1 & -\text{Sin}\left[\frac{\pi }{14}\right] & -\text{Cos}\left[\frac{\pi }{7}\right] & \text{Sin}\left[\frac{3 \pi }{14}\right] & \text{Sin}\left[\frac{3 \pi }{14}\right] & -\text{Cos}\left[\frac{\pi }{7}\right] & -\text{Sin}\left[\frac{\pi }{14}\right] \\   1 & -\text{Cos}\left[\frac{\pi }{7}\right] & \text{Sin}\left[\frac{3 \pi }{14}\right] & -\text{Sin}\left[\frac{\pi }{14}\right] & -\text{Sin}\left[\frac{\pi }{14}\right] & \text{Sin}\left[\frac{3 \pi }{14}\right] & -\text{Cos}\left[\frac{\pi }{7}\right] \\   1 & -\text{Cos}\left[\frac{\pi }{7}\right] & \text{Sin}\left[\frac{3 \pi }{14}\right] & -\text{Sin}\left[\frac{\pi }{14}\right] & -\text{Sin}\left[\frac{\pi }{14}\right] & \text{Sin}\left[\frac{3 \pi }{14}\right] & -\text{Cos}\left[\frac{\pi }{7}\right] \\   1 & -\text{Sin}\left[\frac{\pi }{14}\right] & -\text{Cos}\left[\frac{\pi }{7}\right] & \text{Sin}\left[\frac{3 \pi }{14}\right] & \text{Sin}\left[\frac{3 \pi }{14}\right] & -\text{Cos}\left[\frac{\pi }{7}\right] & -\text{Sin}\left[\frac{\pi }{14}\right] \\   1 & \text{Sin}\left[\frac{3 \pi }{14}\right] & -\text{Sin}\left[\frac{\pi }{14}\right] & -\text{Cos}\left[\frac{\pi }{7}\right] & -\text{Cos}\left[\frac{\pi }{7}\right] & -\text{Sin}\left[\frac{\pi }{14}\right] & \text{Sin}\left[\frac{3 \pi }{14}\right]  \end{array}  \right)

Spectrum or frequency domain

\left(  \begin{array}{c}   7. \\   0 \\   0 \\   0 \\   0 \\   0 \\   0  \end{array}  \right)

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