Derivative of convolution

If y = conv(x,w)
What is the derivative of y with respect to w ?
Here y is the convolution between x and w, all are one dimensional. 

Thanks



		
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"reju_vg" <reju_vg@yahoo.com> writes:

> If y = conv(x,w)
> What is the derivative of y with respect to w ?
> Here y is the convolution between x and w, all are one dimensional. 
> 
> Thanks

If y = f(w(t)), then by the chain rule for differentiation,

  dy / dt = (dy / dw) * (dw / dt).

Then just solve for dy / dw:

  dy / dw = (dy / dt) / (dw / dt).

Now that I've answered this homework question for you, why don't you
think twice next time about bypassing the opportunity to think out
your own solution?
-- 
Randy Yates
Sony Ericsson Mobile Communications
Research Triangle Park, NC, USA
randy.yates@sonyericsson.com, 919-472-1124

Hello Reju,

Before you can have a derivative, I have to ask, are y,x,w discrete
time sequences? If they are, then a derivative doesn't make sense
unless you assume some kind of intepolation function is being used.

In this case write each of x and w as convolution sums (I.e., discrete
convolution with the interpolating function - a sinc() perhaps). Thus
you now have two continuous functions x_hat and w_hat. Put these two
functions in a convolution integral and then find the derivative with
respect to w_hat. Finally sample the result to get back to a discrete
sequence. Is that what you needed? 

Clay

Clay wrote:
> Hello Reju,
> 
> Before you can have a derivative, I have to ask, are y,x,w discrete
> time sequences? If they are, then a derivative doesn't make sense
> unless you assume some kind of intepolation function is being used.

A time derivative makes no sense but wrt the coefficients, yes, they do 
make sense.  How do yo derive the LMS algorithm without being able to 
take a derivative wrt the filter coefficients?
> 
> In this case write each of x and w as convolution sums (I.e., discrete
> convolution with the interpolating function - a sinc() perhaps). Thus
> you now have two continuous functions x_hat and w_hat. Put these two
> functions in a convolution integral and then find the derivative with
> respect to w_hat. Finally sample the result to get back to a discrete
> sequence. Is that what you needed? 
> 
> Clay
> 

Stan Pawlukiewicz wrote:
> Clay wrote:
> > Hello Reju,
> >
> > Before you can have a derivative, I have to ask, are y,x,w discrete
> > time sequences? If they are, then a derivative doesn't make sense
> > unless you assume some kind of intepolation function is being used.
>
> A time derivative makes no sense but wrt the coefficients, yes, they
do
> make sense.  How do yo derive the LMS algorithm without being able to

> take a derivative wrt the filter coefficients?
> >
> >

Stan,

In the LMS case there is a continuous error function. Perhaps Reju can
yield moro information about his problem. I.e., is everything discrete
or continuous? Since little info was given, I have to guess at what he
has and wants.

Clay

>Hi All,
Thank you for the reply.

Here the signals are discrete time sequence. To clarify the situation let
me write part of my problem here.
I want to differentiate the equation E{(x1-y2*w21)(x2-y1*w12) with respect
to w21.  Here E{.}is the expectation operator and y2*w21 means the
convolution between y2 and w21. x1, x2, y1, and y2 are the discrete time
speech signals and w12 and w21 are the impulse responses of a room.

Thanks
Reju

>Stan Pawlukiewicz wrote:
>> Clay wrote:
>> > Hello Reju,
>> >
>> > Before you can have a derivative, I have to ask, are y,x,w discrete
>> > time sequences? If they are, then a derivative doesn't make sense
>> > unless you assume some kind of intepolation function is being used.
>>
>> A time derivative makes no sense but wrt the coefficients, yes, they
>do
>> make sense.  How do yo derive the LMS algorithm without being able to
>
>> take a derivative wrt the filter coefficients?
>> >
>> >
>
>Stan,
>
>In the LMS case there is a continuous error function. Perhaps Reju can
>yield moro information about his problem. I.e., is everything discrete
>or continuous? Since little info was given, I have to guess at what he
>has and wants.
>
>Clay
>
>


		
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