FILTERS
This page describes the Doppler D/F filters in greater detail, and provides an explanation for the use of an SCF / LPF combination, instead of a single, resonant - type active filter. ( BPF )



LPF DESIGN EQUATIONS


Here are the design equations for the low pass filters, in case anyone wants to try to re-design the Doppler for a different antenna scan frequency. FYI, the component values in the original design use a Q = 0.707, Fc = 700 Hz, and Ho = 3. As a general rule, the cutoff frequency should be selected so that it lies ( more or less ) midway between the fundamental and first harmonic of the antenna rotation frequency. ( explanation later ) The value of Q will determine how "peaky" the filter response is, at / near the cutoff frequency, ( 0.707 yields the smoothest response ) and Ho represents the passband voltage gain. Beyond the cutoff frequency, the filter "rolls off" at a rate of -12 db / octave. This D/F employs two such stages, so the total rolloff is -24db / octave.



Note that all the units of measure are WHOLE units... Ohms for resistance, Hertz for frequency, and FARADS for capacitance. The value of K cannot EXCEED the quantity on the right side of the equation, but it can be REDUCED, to allow selection of a "convenient" value for C2.


SCF Design Equations


The SCF design equations are very simple.. simple enough to justify an explanation that appeals to intuition. Consider a "special" case of SCF operation, in which the clock frequency equals ZERO... under these circumstances, the SCF circuit reduces to an ordinary RC low pass filter.

For such a filter, the low pass cutoff frequency ( -3db ) represents the frequency at which R = Xc. From algebra, it follows that fc = 1/ ( 2pi RC ). For this argument to be valid, the value of C must equal the sum of ALL the capacitors in the filter, because ALL of the capacitors are used for any signal of non-zero frequency. For frequencies other than zero, the actual bandwidth will be doubled, because the filter is capable of responding to signals which lie below the filterís center frequency, as well as those which lie above it.

There is a design trade - off between SCF bandwidth and filter response time... very narrow filters will require more time to "pump up" or "bleed down" whenever the applied signal "changes". ( i.e. the D/F signal direction changes ) Again, intuition can be applied, because the time required for an RC circuit to "stabilize" after a change of input voltage is related to the RC time constant... after one RC time constant "interval", the voltage will achieve 63 percent of itís "ultimate" value, and after 5 RC time constant intervals, the output voltage will reach 99 percent of itís ultimate value. Slow response can actually be an advantage, especially if a digital readout is employed, which would ( otherwise ) suffer from extreme "jitter" in the readout digits.


AN EXPLANATION


It might seem odd to some folks to employ a Switched Capacitor Filter, followed by multiple sections of low pass filters, to "reconstitute" a Doppler sine wave signal.... why not simply employ a single, highly resonant bandpass filter ? In fact, there are good reasons NOT to do exactly that, and every successful Doppler - type D/F ( that the author has seen ) employs SCFís, in one form or another. Highly resonant bandpass filters can be easily realized with op-amps, but they exhibit some characteristics ( in this particular application ) that make them so "tempermental" that they are practically useless...

The figure below shows the response curves of an simple resonant filter. The amplitude response curve is the familiar "bell" curve that most people know. The phase response curve is probably less familiar, and it represents the the amount of phase shift that occurs to the signal, as it passes throught the filter. Every type of filter exhibits a specific kind of amplitude AND phase response, for various applied frequencies.



The phase response of a filter is generally not important in the context of ordinary ( voice ) communications circuits, because the human ear is relatively insensitive to phase distortion.... a violin will still sound like a violin, even if the phase relationship between the various ( audio ) harmonics of a violin are distorted... only the relative amplitudes of the harmonics will significantly affect the "sound" of a violin.

Resonant filters will have a significant effect on signal phase, unless the driving frequency EXACTLY equals the natural resonant frequency of the filter. In this application, ( Doppler D/Fís ) that causes serious problems, because the phase angle of the signal is important... it is the quantity being measured, to determine signal direction. If a resonant filter ( accidentally ) changes the phase angle of an applied Doppler sine wave, it will also change the indicated signal direction.


CALIBRATION STABILITY


This fact alone wouldnít normally be a problem. If the amount of "accidental" phase change was a specific, constant amount, compensation could be achieved by adjusting the D/F calibration circuit. Unfortunately, there is very little chance that this calibration would remain effective for any significant time interval... slight changes in the filter center frequency and / or slight changes of the Doppler clock frequency would render any calibration useless, very quickly. The Doppler clock would have to be "rock solid" ( crystal controlled ) and the filter would have to be constructed with components that exhibit virtually ZERO sensitivity to temperature, ageing, voltage variations, etc.

The problem is aggravated by very narrow filter bandwidths... For a resonant bandbass filter, the amount of phase shift introduced by the filter equals 45 degrees at the -3db frequencies, and varies from +45 to -45 degrees within the passband. This means the indicated bearings would also vary 45 degrees ( plus or minus ) from the true value, if the frequency "mismatch" between the clock frequency and the filter center frequency varies by an amount equal to 1 / Q.

For a sharply resonant filter, ( example : Q = 100 ) a frequency mismatch of only 1 percent will throw the D/F calibration off by 45 degrees. This is the dilemma... any resonant filter that would be narrow enough to effectively reject voice audio would also be so sensitive to component changes that it would be almost impossible to maintain D/F calibration.


DOPPLER FILTER METHODS


Switched Capacitor Filters donít exhibit this problem because they require an external clock, which "defines" the center frequency of the filter. In Doppler D/Fís, the center frequency of the SCF is always "guaranteed" ( by design ) to EXACTLY equal the driving frequency of the desired waveform. However, thatís not the end of the story.... Switched Capacitor Filters have an interesting quality that makes them different from ordinary resonant filters... They exhibit multiple resonant frequencies, all harmonically related. These harmonics will distort the Doppler sine wave, and must be attenuated to achieve a sine waveform of reasonable quality. For this reason, it is common practice to feed the SCF output to a series of LPF stages, to "roll off" the Doppler harmonics.

Low pass filters also exhibit a phase / frequency relationship which will affect D/F calibration, but most of the phase - frequency "sensitivity" occurs near the cutoff - frequency of the filter.... for signals which do not lie near the LPF cutoff frequency, the amount of phase - frequency "sensitivity" is greatly reduced, so it is possible to maintain D/F calibration with simple circuits. For this reason, it is common ( in Doppler D/Fís ) to design the LPFís so that the cutoff frequency clearly lies above the fundamental Doppler frequency, and cleary below the first harmonic.


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