4. Response Characteristics

    The type of filter design depends on the desired response characteristics. The complexity of the filter design and the number of crystals used in a filter depends on the required shape factor and the attenuation.  Common design types are discussed below.

Bessel or Linear Phase :  The transfer function of the filter is derived from a Bessel polynomial.  It produces filters with a flat delay around center frequency. The more poles used, the wider the flat region extends. The roll-off rate is poor.  This type of filter is close to a Gaussian filter.  It has poor VSWR and loses its maximally flat delay properties at wider bandwidths.

Butterworth: The transfer function of the filter offers maximally flat amplitude.  Selectivity is better than Gaussian or Bessel filters, but at the expense of delay and phase linearity.  For most bandpass designs, the VSWR at center frequency is extremely good.  Butterworth filters are usually the least sensitive to changes in element values.

Chebyshev:  The transfer function of the filter is derived from a Chebychev equal ripple function in the passband only.  These filters offer performance between that of Elliptic function filters and Butterworth filters.  For the majority of applications, this is the preferred filter type since they offer improved selectivity, and the networks obtained by this approximation are the most easily realized.

Elliptic Function:   The passband ripple is similar to the Chebyshev but with greatly improved stopband selectivity due to the addition of finite attenuation peaks.  The network complexity is increased over the Butterworth or Chebyshev, but it still yields practical realizations over nearly the entire operating region.

Gaussian:  The transfer function of the filter is derived from a Gaussian function.  The step and impulse response of a Gaussian filter has zero overshoot.  Rise times and delay are the lowest of the traditional transfer functions.  These characteristics are obtained at the expensive of poor selectivity, high element sensitivity, and a very wide spread of element values.  Gaussian filter is very similar to the Bessel except that the delay has a slight "hump" at center frequency and the rate of roll-off is slower.  Because of the delay response, the ringing characteristics are better than the Bessel.  Realization restrictions also apply to these filters.

Synchronously Tuned:   These filters have the same advantages and disadvantages as the Bessel and Gaussian except that the ringing response is the best of all design types and the roll-off is even slower that the Gaussian. As with the other two types, some realization restrictions apply.

Chebyshev Phase Error:   In this approximation the Chebyshev approximating technique is applied to the phase (delay) over the passband region.  It produces the bell shaped amplitude response similar to a Gaussian or Bessel design and an equiripple phase and delay response.  The selectivity is better than the Bessel or Gaussian.

Gaussian to 6 (or 12) dB:   This approximation has a passband response that follows the Gaussian shape and, at either the 6 or 12 dB point, the response changes and follows the Butterworth characteristic.  The phase, or delay, response is somewhat improved over a strict Butterworth and the attenuation is better than the pure Gaussian and so it is a true compromise type of approximation.  As with all of the filters where there is an attempt to control the phase response, the realization becomes more difficult and so its operating region is slightly restricted.

5. Stability of Crystal Temperature Filters

     There are some crystals filters built below 1 MHz, but today, most are above and use AT-cut crystals.  While coils or other factors can influence the stability, the crystals are the main controlling components, and in some filter such as MCF the only ones.  Therefore, the stability can be linked to the stability of the AT-cut crystals.  The frequency change vs. temperature follows a cubic characteristic as shown below.  The variation of the family of curves is controlled by slight changes in the angle at which the individual crystal blank is cut from the crystal bar.  Each curve is offset from the adjacent one by a change of only two minutes of arc.  The designer will select the proper curve (angle) to give the minimum deviation over the specified temperature range.