[SOLVED] Sampling problem of a multifrequency signal.

[ashish.mw]

Question:
A signal with multiple frequencies as 0.3KHz, 0.4KHz, 1.3KHz, 3.6KHz, 4.3KHz is sampled with a sampling frequency of 2KHz. wen the sampled signal is passed through LPF with a bandwidth of 900 Hz, what would be the resulting frequency components?

 

[ZekeR]

I'm assuming that the LPF is implemented AFTER filtering, rather than being an anti-aliasing filter...

0.3kHz remains same.
0.4kHz remains same.
1.3kHz aliases to 0.7kHz.
3.6kHz aliases to 0.4kHz.
4.3kHz aliases to 0.3kHz.

The final spectrum will contain components at 0.3, 0.4, and 0.7kHz. The LPF will not do anything.

 

[ashish.mw]

The LPF will not do anything.[/QUOTE]

if there is no role of LPF here, then how come the resulting spectrum is limited to only these three frequencies? I have no problem with your reasoning whatsoever for the answer, but your this point that LPF won't do anything is in my sense wrong. Correct me if i am wrong.

Thanks for your reply.

 

[ZekeR]

Since your signal is sampled at 2kHz, the maximum frequency that can theoretically be represented is 1kHz (Nyquist Theorem). A LPF at 900Hz can only filter out anything between 900Hz and 1kHz. The act of sampling sends all of those higher frequencies into the sampled signal by aliasing, and they are now represented as low-frequency image signals (which cannot be subsequently filtered out digitally).

If you want to prevent anything above 900Hz from being aliased into your signal, you need to implement your LPF as an analog filter, BEFORE your sampler (i.e., an anti-aliasing filter).

 

[ashish.mw]

Theoretically speaking:

(say) consider a signal with frequency fm Hz, being sampled at fs Hz. Now the frequency components available at the output should be all:

fm, fs-fm, fs+fm, 2fs-fm, 2fs+fm,....and so on.

be it sampled at Nyquist rate or not, these are the components that would be available.

Now we use the LPF to band limit our output to certain frequency range only.

Is that notion correct?

 

[ZekeR]

Yes. If fs/2 < fm < fs, then the aliased image signal fs-fm will exist within the bandwidth of other, real signals (the region between 0 < f < fs/2).

 

[FvM]

Now we use the LPF to band limit our output to certain frequency range only.

In a sampled signal, spectral comments beyond -fs/2 < f < fs/2 can be neither unequivocally identified nor filtered. (the negative frequency range exists only for "analytical" signals, respectively real signals sampled by quadrature input stage). Low-pass filters in the digital domain are working only for f < fs/2.

 

[ashish.mw]

Low-pass filters in the digital domain are working only for f < fs/2.

i did not get this, please give me the reason why is it so? moreover in a multifrequency signals, as until otherwise specified, we use fs=2fm where fm is the maximum frequency components, then in that case all the frequencies would be available corresponding to all the frequency components.

Please also consider another case: suppose that if the question would have asked for with a LPF of Bandwith 1.5KHz, and then we are required to find out the components at the output, then what all components would be present at the output?

 

[FvM]

You can't implement a 1.5 kHz LPF with 2 kHz sampling frequency.

Instead of a lengthy explanation, see the magnitude response of a 900 Hz digital LPF with 2 kHz sampling frequency. I think, it's rather visual why it's not "working" for frequencies above 1 kHz. If you don't understand why the frequency response is periodically repeated, refer to a digital signal processing text book.

 

[ashish.mw]

Please if you don't mind in explaining me as i am still unclear. Sorry for inconvenience, but i can not understand while plotting the response of Butterworth filter or any other filter, what is the requirement of sampling? Please help as i can feel i am missing a very big thing in terms of concept.

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Also, i have seen questions of undersampling case with Bandpass filters, so what is your view for such questions?

 

[FvM]

i can not understand while plotting the response of Butterworth filter or any other filter, what is the requirement of sampling?

The sampling frequency is a fundamental parameter of a sampled data system, not a requirement for plotting a filter response.

The frequency domain representation of a sampled signal is comprised of the base band and a sequence of image bands shifted by +/-n*fs. It's impossible to determine if a particular sampled data component is a base band or aliased signal after sampling. The only way to avoid this ambiguity is to filter all signals outside the base band before sampling.

For the problem in the intial post, you could e.g. sample the input signal at 16 kHz, apply a digital low pass filter and decimate the sampling rate to 2 kHz.

Undersampling can produce unequivocal digital signals, if the input signal is bandpass filtered. This is e.g. done in digital receivers. In other words, you select a single image band instead of the base band.

 

[ashish.mw]

In my view, the given question has never stated that there is a need to avoid aliasing or undersampling, so no question of considering the filter as anti-aliasing one.

Next, if we view the problem as a simple numerical, then i think what i said that, LPF is majorly used for limiting the sampled output, that is why, it is to be used after sampling and not before it (as in case of anti-aliasing one).

Am i correct?

Also consider, if there is only one frequency in the signal (say sine or cosine) and now if the signal is undersampled then what is your take on how to use LPF as an anti-aliasing filter.

Also please clarify more on the decimation part.
One more thing, how to select Nyquist sampling rate in cae of multi frequency signals; it is just doubling the highest frequency component, now in that case, there would be no undersampling. correct?

Please help.
Thanks in advance.

 

[FvM]

The original post is clear about "the sampled signal is passed through LPF" which refers to a digital filter that has to deal with all the said aliasing and limited baseband problems. An analog anti-aliasing LPF would lead to a completely different problem with a very simple result.

The original post is in fact describing an undersampling case.

One more thing, how to select Nyquist sampling rate in cae of multi frequency signals; it is just doubling the highest frequency component, now in that case, there would be no undersampling. correct?

Yes.

About decimation. It's an option if you want a 2 kHz output sample rate and filtering of the out-of-band frequencies by a digital filter (e.g. 900 Hz LPF). All recent digital audio analog frontends are using this method.

 

[ashish.mw]

Thanks a lot, the problem is clear now and i am marking it solved.