how can I analyze a basic sine wave for its frequency phase and amplitude at a particular point in time in matlab

Question

How can I analyze a basic sine wave for its frequency phase and amplitude at a particular point in time in MATLAB? Are there any tools to do that? I am trying to do additive synthesis of inharmonic sounds in MATLAB, but having a great difficulty to extract the needed phase amplitude and frequency data of the partials I am trying to generate. Can someone shed some light regarding signal analysis in MATLAB?

Answer 1

If your the unknown signal $x(n)$ is modeled as: $$x(n)=A \sin(2 \pi f n+ \phi)+w(n)$$ and you want to estimate $A$,$f$,and $\phi$ accurately, you can use least square estimation. Unfortunately the cost function is nonlinear. You can use nonlinear least square in MATLAB to find the parameters as follows:

Make a cost function:

f=@(A,f,phi) x[n]-A sin(2*pi*  f *n+ phi)

and use

p0=[A0,f0,phi0];

p = lsqnonlin(f,p0);  

to find the unknown parameters. Note that the optimizer will have a hard time finding $f$ as the problem is not convex. So it is best if you can give an initial estimate of frequency by using a method like fft.

If the frequency is known then the problem can be converted to a linear estimation as: $$x(n)=A \sin(2 \pi f n+ \phi)+w(n)$$ $$=A \sin(2 \pi f n) \cos (\phi) +A \cos(2 \pi f n) \sin (\phi)+w(n)$$ $$=p_1 S[n] +p_2 C[n]+w(n),$$ where $p_1=A \cos(\phi)$ and $p_2=A \sin(\phi)$ are unknown parameters, and $S[n]$ and $C[n]$ are known.

Answer 2

Here's some Matlab code:

% Assume 'x' is the input time-domain sequence and 'y' is it's Hilbert transform.

Analytic_Signal = hilbert(x);
y = imag(Analytic_Signal);

Instantaneous_Magnitude = abs(Analytic_Signal);

Instantaneous_Phase = unwrap(atan2(y, x));

Instantaneous_Freq = diff(Instantaneous_Phase);

Answer 3

if the sinusoid, $x(t)$, is uncontaminated, i s'pose you can measure all three by first computing the Hilbert transform of it, $\hat{x}(t)$ and, from the two, construct the analytic signal,

$$ x_a(t) \ \triangleq \ x(t) \ + \ j\hat{x}(t) $$

the amplitude will be $|x_a(t)|$, the instantaneous phase will be $\phi(t) = \arg\{x_a(t)\}$ and, if the phase is unwrapped the derivative of the phase w.r.t. time is the instantaneous (angular) frequency is this derivative.

$$ \omega(t) \ = \ \frac{d}{dt}\phi(t) \ = \ \frac{d}{dt}\arg\{x(t) \ + \ j\hat{x}(t)\} $$