# Re: Complex version of an impulse

Started by April 16, 2004
```>Subject: Re: Complex version of an impulse
>From: Jerry Avins jya@ieee.org
>Date: 4/16/2004 11:46 AM Eastern Daylight Time
>Message-id: <407fffe8\$0\$16464\$61fed72c@news.rcn.com>
>
>robert bristow-johnson wrote:
>
>> In article da4d20d8.0404152109.7ed6b813@posting.google.com, Impulse at
>> impulse@e.coolworks.com wrote on 04/16/2004 01:09:
>>
>>
>>>Hi all,
>>>
>>>I've got an analyic signal for which I'm designing an IIR
>>>filter with purely real valued coefficients. I'd like to
>>>look at the impulse response of this filter, but since the
>>>normal impulse is purely real and the coefficients are all
>>>real, the impulse response is also purely real.
>>>
>>>In order to get a complex impulse response, I need a
>>>complex impulse. Is this:
>>>
>>>....  0 0 0 0 0 (1+i) 0 0 0 0 0 0 ....
>>>
>>
>>
>>
>> there is a paper called "the Analytical Impulse" by Andrew Duncan in the
>AES
>> Journal that might suggest:
>>
>>     x[n] = d[n] + j*Hilbert{ d[n] }         ("d[n]" is the discrete
>impulse)
>>
>> as the thing to bang a complex linear system with.
>>
>> i dunno.
>>
>> r b-j
>
>I don't get it. In the real world, an analytic signal has two parts, I
>and Q, that exist on two wires. To filter an analytic signal, the parts
>must be filtered separately. Inside a computer -- or our heads -- there
>is more flexibility, up to a point. The samples of an analytic signal
>may be thought of as pairs of samples (labeled I and Q, e.g.) or as
>complex numbers with real or imaginary parts. Either way, the parts can
>be filtered as separate streams. It seems to me that without great
>cleverness, they have to be.
>
>Jerry
>--
>Engineering is the art of making what you want from things you can get.
>&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
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We have to be careful with the terms used in describing signals. Analytic
signals and complex (I and Q) signals are not necessarily the same. Reference
pages 58-59 of Radar Detection by DiFranco and Rubin. DiFranco and Rubin define
an analytic signal as y(t) = s(t) + jx(t),
where x(t) is the Hilbert transform of the real signal s(t). The Fourier
transform of y(t) has the properties Y(w) = 2S(w) for w>0, Y(0) = S(0), and
Y(w) = 0 w<0. DiFranco and Rubin go on to say that under narrowband and
bandlimited conditions, then y(t) = v(t) exp(j*2*w0*t) where v(t) = a(t)
exp(jp(t)) and s(t) = a(t) cos(w0*t + p(t)). Note in the complex signal
representation I = v(t) cos(w0*t) and Q = j v(t) sin(w0*t).

Scott
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