Using DFT to process 1 sec of data and detect 2 sinusoids

[chvti]

A DSP HW Problem

In a recent article, a mathematician claimed that by using DFT to process 1 second of signal data it was possible to detect the presence of two sinusoids of roughly the same amplitude but separated by only 0.75 Hz. Is that really possible? Give a complete explanation of your answer. (No, you don't need to know the sampling rate!)

I appriciate whoever can give me some help here. Thank you!!

 

[Circuit_seller]

Re: A DSP HW Problem

Hi

The resolution in frequency domain depends on the number of samples
for an rectangular window resolution is Fs /M, so increasing number of samples increase frequency resolution.
you can calculate minimum sampling frequency which enables you to reslove two 0.75Hz seprated signals with 1second time interval samples.

Regards

 

[chvti]

Re: A DSP HW Problem

Thank you

 

[chvti]

Re: A DSP HW Problem

Can you be more specific on "you can calculate minimum sampling frequency which enables you to reslove two 0.75Hz seprated signals with 1second time interval samples?" I am not sure exactly how you do that... Thanks again!

 

[Circuit_seller]

Re: A DSP HW Problem

Hi

I said that for a rectangular window FFT frequency resolution is Fs/M, so if our resolution is 0.75 Fs/M = 0.75.
In the other hand Fs.M ( recording interval) is 1 second. replacing this in above results in fs = sqrt(0.75).

 

[chvti]

Re: A DSP HW Problem

I thought about it again. I have a few questions:

1. We do not need to know the sampling rate in order to solve this probem (that’s what the problem says), but you did from calculating the sampling frequency. Isn’t sampling frequency = sampling rate?

2. Why did you pick rectangular window? My prof. never mentioned that rectangular window has FFT frequency resolution of Fs/M. How did you get that? What about if applying other windows such as Hanning and Hamming?

Personally, DSP is not one of my strong areas. I appreciate your efforts in this. Thank you!!

 

[GeorgeM]

Re: A DSP HW Problem

I think, it is impossible to get the resolution of 0.75 Hz with 1 second window, because bandwidth of spectrum line of sine frequency is 2/T=2Hz, where T is window duration.

To get such resolution T=2/0.75=8/3~2.7s

There are a special ways like a super resolution but that's another tale.

 

[Circuit_seller]

Re: A DSP HW Problem

Hi

You can refer to Oppenheim DSP book which states that for a rectangular window the main lob BW is 2*pi / M which 2*pi means Fs.

Regards

 

[chvti]

Re: A DSP HW Problem

1 second of signal data gives you about 1hz resolution, but they are separated by only .75hz. Thus, they are not resolvable. Does that make sense?

 

[virgorabbit]

Re: A DSP HW Problem

Since you have 1 sec data frame, the sample rate is determinded by how many samples in this 1 sec data frame. If there are 10 samples in 1 sec data frame, the sample rate Fs is 10Hz.
To increase the freq resolution of the data, just pad zeros at the end of the data frame, then do the FFT, the FFT point value N should be equal to the data frame length plus the paded zeros. Freq resolution = Fs/N. Larger the N is , higher the freq resolution.

 

[Circuit_seller]

Re: A DSP HW Problem

Hi

I am agree with you.
We can increase frequency resolution both by increasing sampling frequency and zero padding.