How do you get (7) from (5)? I can't deal with these things.

>>8794862
Is this about electricity ? I see things about filters and transfer functions. If so, this is probably a thesis paper, and not very helpful, since you can solve the problems much more easily.

>>8794880

It is about DSP. I want to design a fractional delay using that method, because I know for the graphics on the next page that it is a good one.

Problem is I can't just drop the formulas on my thesis, I need to explain them. But I can't, because I am not able to understand what they are doing.

>>8794884
I'm not smart enough to help you there. I only know basic circuitry.

What is the difference between "fractional delay" and a simple temporal phase ?

>>8794904

Imagine you want to delay a signal 16 samples. No problems, you save a sample and drop it 16 samples later.

But if you want to delay a signal 16,5 samples, that's a problem, because in digital signals there is no sample between 16 and 17. Therefore you have to interpolate the samples around to "guess" the value of that point.

What are doing in that paper is taking the frequency response of the ideal filter that delays the signal a number of samples D plus a fracional part p (look at expression 4) and then minimize the function (5) to obtain a matrix A and substitute it on expression (1).

Once you have that you can implement the system using an efficient structure called the "Farrow structure".

Sorry for my broken english.

(3) states that H (w,p)=w^tap and (5) states that
J (A)=int [0,pi]int [0,1]dpdw W (w,p)|H (w,p)-Hd (w,p)|^2.

|H (w,p)-Hd (w,p)|^2=
|H (w,p)|^2-2|H (w,p)Hd* (w,p)|+|Hd (w,p|^2

|H (w,p)|^2=H×H*=w^tApp^yaw where H* is the complex conjugate if H.

|Hd (w,p)|^2=Hd×Hd*=e^(-jw (D+p))×e^(jw(D+p))=1

2|H×Hd*|=2Re [w^tApe^(jw (D+p))]

Substitute the values of these functions into (5) to extract (7).

>>8794982
|H (w,p)|^2

Holy shit, I forgot about the absolute values....Thanks man....I am so dumb.

>>8795006
No prob

>>8794862
>>8794982
>>8795702

I managed to go through this and it is pretty brilliant, at least for me.

How can someone come up with that? I guess it is only possible after a lot of practice.

Tell me if someone is interested in the whole paper.

>>8796338
I'd like to read the paper if you have a link.

>>8796883
I uploaded it here

https://www.docdroid.net/OXeqBh9/documentsmx-an-improved-weighted-least-squares-design-for-variable-fractional-delay-fir.pdf.html

There is something confusing for me. When he uses the property of the matrix trace at (8), why does he use the Re operator with the second member?

I guess it makes sense if you expand the expression.

I do not really understand the demonstration of the appendix either. But at least I got the rest of the paper.

>>8796927
If one assumes the weighting function W1(w) is real then the product of w^T and it's complex conjugate should yield one since w is defined as an element of the vector (1,e^-jw,...,e^-jnw).

The transpose of w^T should be an nX1 matrix and it's complex conjugate is a 1Xn matrix. Their product yields a 1X1 matrix that is purely real.

Let f (z)=g (z)*h (z) be a function admitting complex numbers z=x+in. If the imaginary part of
f (z) equals 0 as I think the second line of 8 does then:

f (z)=Re [f (z)].

Thanks for the link.

>>8796927
I wasn't able to download the paper, so I can't comment on the appendix. Sorry about that.

>>8797179
>https://www.docdroid.net/OXeqBh9/documentsmx-an-improved-weighted-least-squares-design-for-variable-fractional-delay-fir.pdf.html

You can download it if you click on the upper right button called "download".

>>8797175

> If the imaginary part of f (z) equals 0 as I think the second line of 8 does

In fact, the evaluation of OMEGA at (11) only gives that result if you use the Re operator.

Check it. The product of the complex conjugate of w and w^T is a Toeplitz matrix where o(i,k) = e^jw(i-k) , therefore the integral has an imaginary part.

Perhaps you multiplied them the other way around, or perhaps I didn't understand you well.

Interesting. I can't say I know exactly why the real part is taken in 8; perhaps it's a consequence of w^T's exponential property since the modulus squared of an exponential function is purely real.

If it isn't that, then beats me.

>>8797321
>>8797389

So nobody can tell why? One last bump