Waves and Fourier sums |
In theoretical work and in programs, the unit delay operator definition is often simplified to , leaving us with . How do we know whether is given in radians per second or radians per sample? We may not invoke a cosine or an exponential unless the argument has no physical dimensions. So where we see without , we know it is in units of radians per sample.
In practical work, frequency is typically given in cycles/sec or Hertz, , rather than radians, (where ). Here we will now switch to . We will design a computer mesh on a physical object (such as a waveform or a function of space). We often take the mesh to begin at , and continue till the end of the object, so the time range . Then we decide how many points we want to use. This will be the used in the discrete Fourier-transform program. Dividing the range by the number gives a mesh interval .
Now let us see what this choice implies in the frequency domain. We customarily take the maximum frequency to be the Nyquist, either Hz or radians/sec. The frequency range goes from to . In summary:
What if we want to increase the frequency resolution?
Then we need to choose larger than required to
cover our object of interest.
Thus we either record data over a larger range,
or we assert that such measurements would be zero.
Three equations summarize the facts:
Increasing range in the time domain increases resolution in the frequency domain and vice versa. Increasing resolution in one domain does not increase resolution in the other. |
Waves and Fourier sums |