Jerome,

I saw the scripts so at least your convolve works for (1 + z^-1)^N and 
generates
Pascal's triangle.  Now try (1 + 0.5z^-1)^N for 1<=N<=6 and check by hand.
If this works you can be pretty sure your convolve is working.


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Subject: Project Question

>Should we use the built-in sinc funtion in Matlab or sin(x)/x?
>This also leads to another question:  Is the sinc function defined as
>sin(x)/x or sin(pi*x)/(pi*x)?  According to Communication EE546's
>book and Matlab, sinc(x) is defined as sin(pi*x)/(pi*x).

At least one of my DSP books defines sinc(x) = sin(x)/x (call this
definition #1).  At least one of my DSP books defines sinc(x) =
sin(pi*x)/(pi*x) (call this definition #2).  All my communication books use
definition #2.  I was taught with definition #1 so that's why I presented
it to you in this way.  All my algorithms assume definition #1.

The advantage of definition #2 is that with the right choice of x, the zero
crossings are at integer values.  On the other hand definition #1 seems
more natural so we don't have to carry pi around.  Also the ideal LPF can
be written quicker and in slightly more compact form using definition #1.

For example, on p. 54 of Ludeman, (1.25) gives the impulse response of the
ideal LPF with cutoff omega_c as:

        h(n) = sin(omega_c n)/(pi n)

Using definition #1 this is equivalent to:

        h(n) = (omega_c/pi) sinc(omega_c n).

Using definition #2 this is equivalent to:

        h(n) = (omega_c/pi) sinc(omega_c n / pi).

The former is slightly more compact and easier to attain.  My final advice
(after this thesis): stick to one definition (I don't care which) just be
sure you use it consistantly and correctly.


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Q: I have  questions about last problem.First why do we need to use HPF? I 
think we only need to use LPF and BPF. Second what is the purpose for 
"MODULATE"? What is w0? Finally how can we get gain of a filter? Thank you for 
help.

A: The hearing aid needs a HPF for high frequency compensation.  If you can
compensate without a HPF be my guest but you'll need to prove to me your idea
is sound.  Since all filters are bandpass (LPF = BPF with w1 = 0, w2 = wc;
HPF = BPF with w1 = wc, w2 = pi) this might be what you really mean instead.
In any event you need modulate to construct the BPF from LPFs.  Remember, we
can't use frequency_shift since the non-uniform frequency bands would mean 
complex output.  Finally, I explicity show how to set the gain of the filter
towards the end of the problem.
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