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

syms x;
f = cos(x);
H = ihtrans(f)

H = $$-\sin \left(t\right)$$

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

syms H(t) f(s);
H(t) = sinc(t);
f(s) = ihtrans(H,s)

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

Plot the sinc function and its inverse Hilbert transform.

fplot(H(t),[0 6],'b')
hold on
fplot(f(s),[0 6],'r')
legend('sinc(t)','f(s)')

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

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

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

Apply a 90-degree phase shift to the positive frequency component using the inverse Hilbert transform. Specify the independent variable as x and the transformation variable as u, respectively.

f = ihtrans(H,x,u)

f = $$A\hspace{0.17em}\cos \left(5\hspace{0.17em}u+20\hspace{0.17em}\pi \hspace{0.17em}t\right)$$

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

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

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

f = ihtrans(Z)

f = $$-A\hspace{0.17em}{\mathrm{e}}^{-10\hspace{0.17em}u\hspace{0.17em}\mathrm{i}}\hspace{0.17em}\mathrm{i}$$

Create a real-valued signal $$f(s)$$ with two frequency components, 60 Hz and 90 Hz.

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

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

Calculate the corresponding analytic signal $$F(t)$$ using the inverse Hilbert transform.

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

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

Calculate the instantaneous frequency of $$F(t)$$ using

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

InstantFreq(t) = diff(angle(F(t)),t)/(2*pi);
assume(t,'real')
simplify(InstantFreq(t))

ans = $$75$$