Conservation of Energy

Let us check whether the Law of Conservation of Energy holds: the total power in the time domain is equal to the total power in the frequency domain.

As shown in Appendix D, the power of a sinusoidal signal is given by:

\[ P = \frac{A^2}{2} \]

whereas the power of a DC signal is:

\[ P = A^2 \]

For DFT frequency components (\( k > 0 \)):

\[ P_k = \frac{A_k^2}{2} = \frac{a_k^2 + b_k^2}{2} \]

For the DC component (\( k = 0 \)):

\[ P_0 = A_0^2 = \left(\frac{a_0}{2}\right)^2 = \frac{a_0^2 + b_0^2}{4} \]

Let us calculate the total power of the spectrum for our example. The total power \( P_X \) is given by:

\[ P_X = P_0 + \sum_{k=1}^{N-1} P_k \]

For DFT frequency components (\( k > 0 \)), each frequency component appears twice - once for the positive frequency and once for the corresponding negative frequency (due to the As shown in symmetry of the real DFT). Therefore, to avoid double counting, the power \( P_k \) should be halved:

\[ P_X = P_0 + \frac{1}{2} \sum_{k=1}^{N-1} P_k \]

Expressing this in terms of the Fourier coefficients:

\[ P_X = \frac{a_0^2 + b_0^2}{4} + \frac{1}{2} \sum_{k=1}^{N-1} \frac{a_k^2 + b_k^2}{2} = \frac{1}{4} \sum_{k=0}^{N-1} (a_k^2 + b_k^2) \]

Substituting the numerical values:

\[ P_X = \frac{1}{4} \left( 2^2 + 3^2 + 0^2 + 2^2 + 0^2 + 2^2 + 0^2 + 3^2 \right) = 7.5 \]

Now, let us compare this result to the power of the sampled input signal, given by:

\[ P_x = \frac{1}{N} \sum_{n=0}^{N-1} x(n)^2 \]

Substituting the values:

\[ P_x = \frac{1}{8} \left( \left(\frac{3-\sqrt{2}}{\sqrt{2}}\right)^2 + \left(\frac{2+2\sqrt{2}}{\sqrt{2}}\right)^2 + \left(\frac{3-3\sqrt{2}}{\sqrt{2}}\right)^2 + \left(\frac{2-\sqrt{2}}{\sqrt{2}}\right)^2 + \left(\frac{-3-\sqrt{2}}{\sqrt{2}}\right)^2 + \left(\frac{-2-4\sqrt{2}}{\sqrt{2}}\right)^2 + \left(\frac{-3+\sqrt{2}}{\sqrt{2}}\right)^2 + \left(\frac{-2-\sqrt{2}}{\sqrt{2}}\right)^2 \right) = 7.5 \]

Since \( P_X = P_x \), we see that the Law of Conservation of Energy holds: the total power in the time domain is equal to the total power in the frequency domain.

The Parseval’s Theorem

Parseval's theorem states that a signal's total energy (or power) in the time domain is equal to the total energy (or power) in the frequency domain.

For a discrete-time signal \( x(n) \) of length \( N \), Parseval’s theorem states:

\[ \sum_{n=0}^{N-1} |x(n)|^2 = \frac{1}{N} \sum_{k=0}^{N-1} |X(k)|^2 \]

This equation applies to the unnormalized DFT. If we normalize the DFT by a factor of \( \frac{2}{N} \), we need to adjust the power relation accordingly.

\[ X_{\text{norm}}(k) = \frac{N}{2} X(k) \]

Substituting into Parseval’s theorem:

\[ \sum_{n=0}^{N-1} |x(n)|^2 = \frac{1}{N} \sum_{k=0}^{N-1} \left| \frac{N}{2} X(k) \right|^2 \]

\[ = \frac{1}{N} \sum_{k=0}^{N-1} \frac{N^2}{4} |X(k)|^2 = \frac{N}{4} \sum_{k=0}^{N-1} |X(k)|^2 \]

Dividing both sides by \( N \):

\[ \frac{1}{N} \sum_{n=0}^{N-1} |x(n)|^2 = \frac{1}{4} \sum_{k=0}^{N-1} |X(k)|^2 \]