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![]() | Waves and Fourier sums | ![]() |
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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. |
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![]() | Waves and Fourier sums | ![]() |
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