In Problem 4.36 of DSP-Proakis [1], the task is to provide insights into harmonic distortion which may be present in practical sinusoidal generators. Consider the signal

, where .

My take:

The discrete time signal of fundamental period can consist of frequency components separated by radians or cycles (Refer Section4.2 in [1]).

The Fourier series coefficient at frequency is,

.

The total power of the signal over a period is same as the sum of the power of all the Fourier series coefficients (harmonic components) i.e.

.

As the signal is real valued, the power at other frequency components apart from the desired frequency component is .

Concluding, the Total Harmonic Distortion (THD) is

.

In the second part of the problem, the task is to generate the sinusoidal for a single period by Taylor approximation

.

f = 1/32;

t = repmat([0:(1/f)-1],1,1);

x = 2*pi*f*t;

n = [0:7]’;

s = factorial(2*n);

p = (-1).^n;

r = (p./s*ones(size(t))).*(ones(size(n))*x).^(2*n*ones(size(t)));

cosT = sum(r); % cosine computed from Taylor series

cosA = cos(x); % actual cosine values

The third part of the problem is to find the THD of the sinusoidals obtained from the Taylor series summation and the actual cosine function.

% compute the fourier series coefficients

k = [0:(1/f)-2]’;

refF = exp(-j*2*pi*k*t*f);

hdA = mean((ones(1/f -1,1)*cosA).*refF,2);

hdT = mean((ones(1/f -1,1)*cosT).*refF,2);

% THD from Taylor series based cosine function

THD_T = 1 – 2*abs(hdT(2)).^2/sum(cosT.^2/length(cosT))

% THD from actual cosine function

THD_A = 1 – 2*abs(hdA(2)).^2/sum(cosA.^2/length(cosA))

From a quick run of the code for different values of f, observed that the computed THD for lower frequencies is higher than that for higher frequencies. For a lower frequency signal, there are more harmonic components present in , hence the reason.

Also, as only a single period of the sinusoidal signal is considered, the signal can be treated as an aperiodic signal and hence having a continuous (and periodic) spectrum. Summarizing, apart from the power at the harmonic frequencies, the signal will be having frequency components at other portions of the spectra in .

References:

[1] Digital Signal Processing – Principles, Algorithms and Applications, John G. Proakis, Dimitris G. Manolakis