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Calculating the energy and power of a signal was discussed in one of the previous posts. Here, we will verify the method by using Discrete Fourier Transform (DFT) in Matlab. Check here to know more on the concept of power and energy.

The total power of a signal can be computed using the following equation
$$P_x=\lim_{N\rightarrow \infty }\frac{1}{N}\sum_{n=0}^{n=N-1}\left | x(n) \right |^2 \rightarrow (1) $$

For other forms of equations: refer here.

Case Study:

\(x(t)\) is a sine wave of amplitude \(A\) and frequency \(f_c\) represented by the following equation.
$$ x(t) = A sin(2 \pi f_c t) $$
When represented in frequency domain, it will look like the one on the right side plot in the following figure. This is evident from the fact that the sinewave can be mathematically represented by applying Euler’s formula.
$$ Asin(2 \pi f_c t) = A\frac{e^{j 2 \pi f_c t}-e^{-j 2 \pi f_c t}}{2j} $$

Taking the Fourier transform of \(x(t)\) to represent it in frequency domain,
$$X(f)=F \left[Asin(2 \pi f_c t) \right]=\int_{-\infty }^{\infty }\left[\frac{e^{j 2 \pi f_c t}-e^{-j 2 \pi f_c t}}{2j}\right]e^{-j2 \pi f t} dt = \frac{A}{2j}\left[ \delta (f-f_c) - \delta (f+f_c) \right ]$$

When considering the amplitude part,the above decomposition gives two spikes of amplitude A/2 on either side of the frequency domain at \(f_c\) and \(-f_c\).

Time domain representation and frequency spectrum of a pure sinewave
Squaring the amplitudes gives the magnitude of power of the individual spikes/frequency components. The power spectrum is plotted below.

Power spectrum of a pure sinewave
Thus if the pure sinewave is of amplitude A=1V and frequency=100Hz, the power spectrum will have two spikes of value \(A^2/4=0.25 \) at 100 Hz and -100 Hz frequencies. The total power will be \(A^2/4+A^2/4=0.25+0.25=0.5 W \)

Let’s verify this through simulation.

Simulation and Verification

A sine wave of 100 Hz frequency and amplitude 1V is taken for the experiment.

A sinusoidal wave of 10 cycles is plotted here

Sine wave generated in Matlab
 

Matlab’s Norm function:

Matlab’s basic installation comes with “norm” function. The p-norm in Matlab is computed as
$$NORM(v,p) = \left( \sum_{n=0}^{n=N-1}\left | x(v) \right |^p \right)^{1/p} \rightarrow (2)$$

By default, the single argument norm function computed 2-norm given as
$$NORM(v) =NORM(v,2)= \left( \sum_{n=0}^{n=N-1}\left | x(v) \right |^2 \right)^{1/2} \rightarrow (3)$$

To compute the total power of the signal \(x[n]\) (as in equation (1) above), all we have to do is – compute norm(x), square it and divide by the length of the signal.

The above given piece of code will result in the following output

Verifying the total Power by DFT : Frequency Domain

Here, the total power is verified by applying DFT on the sinusoidal sequence. The sinusoidal sequence \(x[n]\) is represented in frequency domain \(X[f]\) using Matlab’s FFT function. The power associated with each frequency point is computed as

$$P_x[f]= X[f]X^{*}[f]$$

Finally, the total power is calculated as the sum of all the points in the frequency domain representation.

Result:

Power spectrum of a pure sinewave simulated in Matlab
A word on Matlab’s FFT: Matlab’s FFT is optimized for faster performance if the transform length is a power of 2. The following snippet of code simply calls “fft” without the transform length. In this case, the window length and the transform length are the same. This is to simplify the calculation of power. You can re-write the call to the FFT routine with transform length set to next power of two which is greater than or equal to the window length (sequence length). Then the step to compute total power will be differing slightly in the denominator. But that will not improve resolution (Remember : zero padding to compute FFT will not improve resolution).

Also note that in the above simulation we are using a pure sinusoid. The entire sequence of sinusoid defined all the cycles completely. There are no discontinuities in the sequence. If you call FFT with the transform length set to next power of 2 (as given in Matlab manuals), it will pad additional zeros to the sequence and creates discontinuities in the FFT computation. This will lead to spectral leakage. FFT and spectral leakage is discussed here.

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