Compute the Hilbert transform of sin(t). By default, the transform returns a function of x.

syms t;
f = sin(t);
H = htrans(f)

H = $$-\cos \left(x\right)$$

Compute the Hilbert transform of the sinc(x) function, which is equal to sin(pi*x)/(pi*x). Express the result as a function of u.

syms f(x) H(u);
f(x) = sinc(x);
H(u) = htrans(f,u)

H(u) = 
$$-\frac{\frac{\cos \left(\pi \hspace{0.17em}u\right)}{u}-\frac{1}{u}}{\pi }$$

Plot the sinc function and its Hilbert transform.

fplot(f(x),[0 6])
hold on
fplot(H(u),[0 6])
legend('sinc(x)','H(u)')

Create a sine wave with a positive frequency in real space.

syms A x t u;
assume([x t],'real')
y = A*sin(2*pi*10*t + 5*x)

y = $$A\hspace{0.17em}\sin \left(5\hspace{0.17em}x+20\hspace{0.17em}\pi \hspace{0.17em}t\right)$$

Apply a –90-degree phase shift to the positive frequency component using the Hilbert transform. Specify the independent variable as t and the transformation variable as u.

H = htrans(y,t,u)

H = $$-A\hspace{0.17em}\cos \left(5\hspace{0.17em}x+20\hspace{0.17em}\pi \hspace{0.17em}u\right)$$

Now create a complex signal with negative frequency. Apply a 90-degree phase shift to the negative frequency component using the Hilbert transform.

z = A*exp(-1i*10*t)

z = $$A\hspace{0.17em}{\mathrm{e}}^{-10\hspace{0.17em}t\hspace{0.17em}\mathrm{i}}$$

H = htrans(z)

H = $$A\hspace{0.17em}{\mathrm{e}}^{-10\hspace{0.17em}x\hspace{0.17em}\mathrm{i}}\hspace{0.17em}\mathrm{i}$$

Create a real-valued signal $\mathit{f}\left(\mathit{t}\right)$ with two frequency components, 60 Hz and 90 Hz.

syms t f(t) F(s)
f(t) = sin(2*pi*60*t) + sin(2*pi*90*t)

f(t) = $$\sin \left(120\hspace{0.17em}\pi \hspace{0.17em}t\right)+\sin \left(180\hspace{0.17em}\pi \hspace{0.17em}t\right)$$

Calculate the corresponding analytic signal $\mathit{F}\left(\mathit{s}\right)$ using the Hilbert transform.

F(s) = f(s) + 1i*htrans(f(t),s)

F(s) = $$\sin \left(120\hspace{0.17em}\pi \hspace{0.17em}s\right)+\sin \left(180\hspace{0.17em}\pi \hspace{0.17em}s\right)-\cos \left(120\hspace{0.17em}\pi \hspace{0.17em}s\right)\hspace{0.17em}\mathrm{i}-\cos \left(180\hspace{0.17em}\pi \hspace{0.17em}s\right)\hspace{0.17em}\mathrm{i}$$

Calculate the instantaneous frequency of $\mathit{F}\left(\mathit{s}\right)$ using

where $$\varphi \left(s\right)=\arg [F(s)]$$ is the instantaneous phase of the analytic signal.

InstantFreq(s) = diff(angle(F(s)),s)/(2*pi);
assume(s,'real')
simplify(InstantFreq(s))

ans = $$75$$