Fourier transform and power spectrum?

Question

I know that we can use fourier transform calculate the power spectrum. If I don't want any signal below 60Hz, can I "zeros" some data from the fourier transform to create an idea high pass filter?

Here is what I mean

complex number array .* real number array = power spectrum. Assume each bin is represent 1 Hz width. Can I "zero" the first 60 bins in the complex number array and real number array, and then do the inverse fourier transfrom? does the result equal to an ideal high pass filter?/

Answer 1

What you describe is termed a "Brick Wall" filter, because it stops all the signals in the stopband as if they had run into a brick wall.

The down side to using zeros in the stopband of your filter is that  the abrupt change in the frequency domain will result in nonlinearities (ringing) in the time domain.  

The better thing to do is apply a Hamming or Hanning window function on the basic rectangular filter to produce some very small non-zero coefficients in the stopband, and the desired gain of very close to 1 in the passband.

For calculating a power spectrum, you square the output of the Fourier transform.  Depending on the coefficients, you may need to apply a 1/2 to the output to obtain a power-invariant transformation.  (Power invariant means it has the same calculated power in the time domain as in the frequency domain.)

Do not just arbitrarily multiply a real array times the Fourier transform output complex array and then think this is a 'power' spectrum.  All you've done is implemented some arbitrary filter.

For an ideal high pass filter, beware of the sampling rate, and note that if the input signal is not properly low-pass filtered prior to sampling, these higher frequencies will wrap around ('alias') into the low frequency spectrum of your Fourier transform.

Good luck!

Answer 2

Yes, but you will never be able to implement it in real life, it would be too high of an order. Is this discrete or continuous time?