Mathematical Sine-Wave Analysis

The above method of finding the frequency response involves physically
measuring the amplitude and phase response for input sinusoids of
every frequency. While this basic idea may be practical for a real
black box at a selected set of frequencies, it is hardly useful for
filter design. Ideally, we wish to arrive at a *mathematical
formula* for the frequency response of the filter given by
Eq.
(1.1). There are several ways of doing this. The first we
consider is exactly analogous to the sine-wave analysis procedure
given above.

Assuming Eq. (1.1) to be a linear time-invariant filter specification (which it is), let's take a few points in the frequency response by analytically ``plugging in'' sinusoids at a few different frequencies. Two graphs are required to fully represent the frequency response: the amplitude response (gain versus frequency) and phase response (phase shift versus frequency).

The frequency 0 Hz (often called *dc*, for *direct current*)
is always comparatively easy to handle when we analyze a filter. Since
plugging in a sinusoid means setting
,
by setting
, we obtain
for all
. The input signal, then, is the same number
over and over again for each sample. It should be clear that
the filter output will be
for all
. Thus, the gain at frequency
is 2, which we get by dividing
, the output amplitude, by
, the input amplitude.

Phase has no effect at
Hz because it merely shifts a constant
function to the left or right. In cases such as this, where the phase
response may be arbitrarily defined, we choose a value which preserves
*continuity*. This means we must analyze at frequencies in a
neighborhood of the arbitrary point and take a limit. We will compute
the phase response at dc later, using different techniques. It is
worth noting, however, that at 0 Hz, the phase of every
*real*^{2.2} linear
time-invariant system is either 0 or
, with the phase
corresponding to a sign change. The phase of a *complex filter*
at dc may of course take on any value in
.

The next easiest frequency to look at is half the sampling rate,
. In this case, using basic trigonometry (see §A.2), we can
simplify the input
as follows:

(2.2) |

where the beginning of time was arbitrarily set at . Now with this input, the

(2.3) |

The filter of Eq. (1.1) thus has a gain of 0 at . Again the phase is not measurable, since the output signal is identically zero. We will again need to extrapolate the phase response from surrounding frequencies (which will be done in §7.6.1).

If we back off a bit, the above results for the amplitude response are obvious without any calculations. The filter is equivalent (except for a factor of 2) to a simple two-point average, . Averaging adjacent samples in a signal is intuitively a low-pass filter because at low frequencies the sample amplitudes change slowly, so that the average of two neighboring samples is very close to either sample, while at high frequencies the adjacent samples tend to have opposite sign and to cancel out when added. The two extremes are frequency 0 Hz, at which the averaging has no effect, and half the sampling rate where the samples alternate in sign and exactly add to 0.

We are beginning to see that Eq.
(1.1) may be a low-pass filter
after all, since we found a boost of about 6 dB at the lowest
frequency and a null at the highest frequency. (A gain of 2 may be
expressed in decibels as
dB, and a
*null* or *notch* is another
term for a gain of 0 at a single frequency.) Of course, we tried only
two out of an infinite number of possible frequencies.

Let's go for broke and plug the general sinusoid into Eq. (1.1), confident that a table of trigonometry identities will see us through (after all, this is the simplest filter there is, right?). To set the input signal to a completely arbitrary sinusoid at amplitude , phase , and frequency Hz, we let . The output is then given by

This input can be simplified as follows: Recall from the discussion surrounding Fig.1.6 that only the peak-amplitude

where , as discussed in §A.1. (The symbol means ``is defined as''.) With this input, the output of the simple low-pass filter is given by

All that remains is to reduce the above expression to a single sinusoid with some frequency-dependent amplitude and phase. We do this first by using standard trigonometric identities [2] in order to avoid introducing complex numbers. Next, a much ``easier'' derivation using complex numbers will be given.

Note that a sum of sinusoids at the same frequency, but possibly
different phase and amplitude, can always be expressed as a
*single* sinusoid at that frequency with some resultant phase and
amplitude. While we find this result by direct derivation in working
out our simple example, the general case is derived in §A.3
for completeness.

We have

(2.4) |

where and . We are looking for an answer of the form

where is the filter amplitude response and is the phase response. This may be expanded as

Therefore,

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University