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Hilbert transform and down sampling

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gongdori

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Hello,

I am a DSP newbie and have question about downsampling after Hilbert transform.

I have a real signal coming into the system and i am going to use Hilbert transform FIR filter to make analytic signal.
Then, since I have only one sided spectrum, I am going to down sample it by 2.

And, here is the question.
I thought I always need to move the spectrum to the baseband before I decimate it. But, I've heard that the location of the spectrum is not important as long as the bandwidth of the spectrum is within the Nyquist criterion... And that puzzled me.
If we have a spectrum which starts from 0 and ends at fs/2 and if we downsample it by 2, how come we don't get any aliasing from the spectrum from fs/4 to fs/2? Aren't they going to be overlapped on the top of the spectrum from 0 to fs/4?

Thanks

Thomas
 

Your question would be clearer if you mention the actual signal specification.

A signal spreading over 0 to fs/2 can't be further decimated without aliasing into the signal band. But I don't understand how the example is related to the previous question or baseband versus passband signal?
 

Thank you for your reply.

Let me try to explain better.
Assuming my spectrum from real signal is spread from -fs/2 to fs/2, when I get rid of one of the two sides of a spectrum by Hilbert Transform, It can be down sampled by 2, I think.
I was not sure how it can be done without shifting the remaining spectrum, from 0 to fs/2, to baseband (shift by fs/4).

My colleague pointed me to chapter 8.2.1 of "Multirate signal Processing for communication system" by Fredric Harris. It talks about how half-band filter centered on baseband can be shifted up to cover positive frequency (0 to fs/2) and can be used as a Hilbert transform filter. However, 2:1 downsampler follows immediately after the filter. I don't understand how signal can be down sampled right after the Hilbert transform without worrying about aliasing.

Thanks!

Thomas
 

I can't tell you about the exact prerequisites, but I assume it's basically possible.

You however need to perform a frequency shift before downsampling, but you didn't.

Apart from the right method for the operation, what's the purpose? You get a complex signal of half sample rate instead of a real one. What do you intend with it. Any practical benefits?
 

Thanks again for your reply.

Yes, there are benefits by doing so.
In my application, I get two consecutive samples in parallel. By reducing the sampling rate by 2, I can deal with one stream of data in downstream.
Also, I need to have complex signal anyway for downstream DSP blocks.

Thanks,

Thomas
 

But the frequency shifted signal is a bandpass signal with a negative carrier frequency. Which DSP operations can be easily applied to it?
 

If you use oversampling as a mechanism of down sampling it will not compromise the Nyquist criteria
 

All,
Thanks again.

I have a bunch of FFTs and other blocks which expect to see complex signal in downstream.
Yes, I agree that band pass signal can be used. however, if the center frequency of the band pass signal is at fs/4 in my example, it will suffer from aliasing, if I downsample it by 2, right?

Thomas
 

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