Hilbert Transform usage in Envelope Detection

[anishsingh]

Hi
I am working on envelope detection of a speech signal using Hilbert transform in matlab.. It's mentioned that assuming signal is y
abs(y + j*hilbert(y) gives envelope of the signal..

How can this be interpreted intuitively?? I didn't understand the logic behind this... Could someone explain this to me

 

[pancho_hideboo]

See "Applications" in https://en.wikipedia.org/wiki/Analytic_signal

 

[v_suma]

For example, y(t) = A cos(w_n * t)
Hilbert( y(t) ) => applying -90 degree phase shift (exp(-j*pi/2)) to the positive frequency components and +90 degree phase shift (exp(j*pi/2)) to the negative frequency components.

y(t) can be rewritten as, y(t) = A cos(w_n * t)
= (A/2) [exp( j * w_n * t ) + exp ( -j * w_n * t )]

Hilbert => (A/2) [exp( j * w_n * t ) exp( -j*pi/2 ) + exp( -j * w_n * t ) exp( +j*pi/2 ) ]
= (A/2) [-j * exp( j * w_n * t ) + j * exp(- j * w_n * t )]
= (A/2) [2 sin(w_n * t )]
= A sin( w_n * t )

Now, y(t) + j Hilbert(y(t)) = A cos( w_n * t ) + j A sin( w_n * t )
= A [cos( w_n * t ) + j sin( w_n * t )]
= A exp( j * w_n * t)

If you take the absolute value of the above expression => A since absolute value of exp is 1.
=> A can be a constant or time varying
=> Envelope is detected.

Hope this helps.