Composite Image Filter - Cascading Sections

Cascading Sections

Several L half-sections may be cascaded to form a composite filter. The most important rule when constructing a composite image filter is that the image impedances must always face an identical impedance; like must always face like. T sections must always face T sections, Π sections must always face Π sections, k-type must always face k-type (or the side of an m-type which has the k-type impedance) and m-type must always face m-type. Furthermore, m-type impedances of different values of m cannot face each other. Nor can sections of any type which have different values of cut-off frequency.

Sections at the beginning and end of the filter are often chosen for their impedance match in to the terminations rather than the shape of their frequency response. For this purpose, m-type sections of m = 0.6 are the most common choice. An alternative is mm'-type sections of m=0.7230 and m'=0.4134 although this type of section is rarely used. While it has several advantages noted below, it has the disadvantages of being more complex and also, if constant k sections are required in the body of the filter, it is then necessary to include m-type sections to interface the mm'-type to the k-types.

The inner sections of the filter are most commonly chosen to be constant k since these produce the greatest stopband attenuation. However, one or two m-type sections might also be included to improve the rate of fall from pass to stopband. A low value of m is chosen for m-types used for this purpose. The lower the value of m, the faster the transition, while at the same time, the stopband attenuation becomes less, increasing the need to use extra k-type sections as well. An advantage of using mm'-types for impedance matching is that these type of end sections will have a fast transition anyway (much more so than m=0.6 m-type) because mm'=0.3 for impedance matching. So the need for sections in the body of the filter to do this may be dispensed with.

Another reason for using m-types in the body of the filter is to place an additional pole of attenuation in the stopband. The frequency of the pole directly depends on the value of m. The smaller the value of m, the closer the pole is to the cut-off frequency. Conversely, a large value of m places the pole further away from cut-off until in the limit when m=1 the pole is at infinity and the response is the same as the k-type section. If a value of m is chosen for this pole which is different from the pole of the end sections it will have the effect of broadening the band of good stopband rejection near to the cut-off frequency. In this way the m-type sections serve to give good stopband rejection near to cut-off and the k-type sections give good stopband rejection far from cut-off. Alternatively, m-type sections can be used in the body of the filter with different values of m if the value found in the end sections is unsuitable. Here again, the mm'-type would have some advantages if used for impedance matching. The mm'-type used for impedance matching places the pole at m=0.3. However, the other half of the impedance matching section needs to be an m-type of m=0.723. This automatically gives a good spread of stopband rejection and as with the steepness of transition issue, use of mm'-type sections may remove the need for additional m-type sections in the body.

Constant resistance sections may also be required, if the filter is being used on a transmission line, to improve the flatness of the passband response. This is necessary because the transmission line response is not usually anywhere near perfectly flat. These sections would normally be placed closest to the line since they present a predictable impedance to the line and also tend to mask the indeterminate impedance of the line from the rest of the filter. There is no issue with matching constant resistance sections to each other even when the sections are operating on totally different frequency bands. All sections can be made to have precisely the same image impedance of a fixed resistance.