About linear transformations

[wasim.nawaz]

Hi everyone,
as it is known that multipliction of two signals in time domain is equivalent to convolution in frequency domain.
moreover
for discrete fourier transform or z-transform etc multiplication in time is equivalent to "circular convolution" in frequency.
Now
for any arbitrary linear transform (say A) with orthonormal basis the above property does not necessarily hold.

if we linearly transform our signals "x1" and "x2" (say A=transformation matrix with orthonormal basis) like

Ax1=y1
Ax2=y2
and
x1*x2=x3 pointwise product of two signals

Ax3=y3

now if i have A, y1 and y3, can i write y2 in terms of A,y1 and y3?
Please respond.
Regards
Wasim

 

[htg]

You assume that A has an orthonormal basis of eigenvectors. So it is invertible.

Use A^-1 (i.e. the inverse of A) to solve the problem.