The problem: We have a digital signal sampled with 600 sample/sec, now we want to decimate the signal to lower sample rates, for example 200, 100, and 50 sample/sec for seismic applications), so the problem here is a combination of aliasing and decimation.
Present solution (example: 600sps to 200sps): designing a digital filter with the desired frequency response and obtain the time domain function of this filter (using inverse FFT, the corresponding SINC function), Then convolve this SINC function with the input signal to obtain the filtered signal with no aliasing.
Questions:
1- what parameters should be evaluated when deciding to choose the best filter design (passband ripple, stopband attenuation, …)? And can we achieve the best design by optimizing all these parameters at the same time, considering our hardware limitations like low sampling rate?
2- Hoe can we obtain the output signal with the minimum attenuation of the input signal in the passband part of the filter, and at the same time the maximum attenuation in the stopband of the filter?
3- What is the best number of order for a lowpass filter? Is sampling rate acts as a limitation here?
Here I have shared an example of using a digital filter ( 600 sample to 200 sample), this filter designed in MATLAB filter builder app, the steps taken are as follows:
1- Type of filter: lowpass
2- Impulse response: FIR
3- Order mode: first try (minimum order or automatic) ----second try (100, 500, and 1000)
4- Filter type: single rate
5- 5- pass band frequency: 90 stopband frequencie: 100 (the desired sample rate is 200sps),
6- In the case of choosing the minimum order design (step 3), the passband ripple was set to be 1dB, and the stopband attenuation to be 80 dB,
7- Design method: Equiripple
8- Structure: direct from FIR
The results are presented in the attached figures,
Now, I would be appreciated if anyone can help me answer the question 1-3 mentioned above, or give me some advice on the procedure, for example how the attenuation of signal in the frequencies lower than 90 Hz, could be avoided?
The tests are carried out by applying a generated signal with the amplitude of 1volt p-p, and at the frequencies of 0.1, 1, 10, 20, 30, 40, 50, 60, 70, 80, 85, 90, 95, 100, 105 Hz to the ADC and obtaining 600 SPS signal, then this digital signal imported in MATLAB.
If you can afford the computational "effort", your current solution (convolution of the input with the signal resulting from the inverse FT of a rectangular window in the frequency domain) is superior to anything else: no aliasing, perfect passband and stopband characteristics. (I've applied a similar algorithm in image processing for sampling rate adaptation and the results are simply impressive.
Mr. Ahmadian,
I suggest you to consider a multirate structure, with the aliasing on the sub-bands cancelling the effects of each other.
Best regards.
Dear Ahmadian,
welcome,
What is the maximum frequency contained in the original analog signal before sampling. This an important information for determining the aliasing limit. If you have the digitized signal you can make on it an fft or dft to sample its frequency content and see the maximum frequency contained in the original analog signal.
Based on this result you can determine lowest sampling frequency to avoid aliasing. You can decimate the sampling frequency to this minimum sampling rate without aliasing.
Best wishes
Dear Ahmadian,
in addition to the previous responses, I may answer:
1/ The parameters are: along X axis, the width of bandpass, bandstop, transition band, in Y axis: ripple and rejection. There is a tradeoff between the lentgh of FIR filter (and then the delay, the complexity) and the quality of ripple and rejection. For instance, you may wish to reject the aliasing, and then you constraint the rejection value. If you constraint the ripple as well, the filter may be very overlong ! However, if you know that relaxed ripple (i.e. "waves in the bandpass") does not disturb your signal, then actually relax the ripple (for instance, if there is an equalization process). Note that there is another parameter for hardware application: the quantization bits you choose per taps. If you are limited, then, it is useless to overconstraint the ripple/rejection. The best method is indeed the equiripple design, as it minimize the minimax cost function, i.e. the design is controlled in the constrained bands.
2/ If you are limited in number of taps, then you have to relax one (or both) of ripple/rejection. There is no general method to optimize this problem, it depends on the use case.
3/ In the same way, there is not a "best" number of taps. Choose the shorter filter that fits your constraints. Note that the sampling rate is not an inherent limitation. However, filter induce delay, and the latency of your system may be another constraint.
Hope this helps,
Vincent
If you can afford the computational "effort", your current solution (convolution of the input with the signal resulting from the inverse FT of a rectangular window in the frequency domain) is superior to anything else: no aliasing, perfect passband and stopband characteristics. (I've applied a similar algorithm in image processing for sampling rate adaptation and the results are simply impressive.
Hello, Hossein,
Unfortunately, I am short on time at the moment; additionally, I can not put my hands on one of my favorite DSP books.
This isn't really a problem of 'low-' vs. 'high-' sampling rates as the techniques are the same, unless you are trying to meet (real-time) throughput constraints - i.e., the sampling rate has to be greater than 2x the max. frequency content.
In general, most of the correctly designed decimation/interpolation/combination algorithms have built-in filters which account for the aliasing problem.
As mentioned previously, you can look at multirate structures. Not the book I was looking for, but "Digital Signal Processing: A computer-based approach", by Sanjit K. Mitra, Ch. 13; specifically, section 13.2 has a nice description. Again, I'm a bit short on time, but basically you want to filter any frequency in the original signal which is above half of the 'new' downsampled/decimation frequency. This gets rid of the aliasing issue, though, you may be throwing out true signal components. Then, you can do the actual decimation/downsampling operation, i.e., keeping every 'N'th sample. MATLAB has both a downsampling and decimation routine.
Regards,
Jim
I want to thank you all for all your good and complete answers,
The Digital filters I have designed and applied on the 600 sps input signal, were low-pass filters with 601 sample length for output 200 sps, and 813 samples length for 50 and 100 sps output, here the problems were computational time for calculation and the attenuation of the input signal in frequencies near the stop band (90Hz for 200sps, 45Hz for 100sps, and 22.5Hz for 50sps),
Now if we could handle the processing time, the attenuation in the pass band of the designed filters remained, and based on the answers above I think I have no choice but to use multirate structures assuming this approach would not need an extra calculation time,
the best results I have obtained with single rate case were (low-pass filters with 600, 800, and 800 order for three needed output sample rates of 200, 100 and 50 sps, the figures attached to this answer,
Thank you all for your support,
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