Question

You are given two bandpass filters The first has a 5-pole Butterworth topolgy with a 10% bandwidth. The second has a 5--pole Chebyshev topology with a 10% bandwidth. Using you knowledge of microwave filters, qualitatively analyze the performance of each and comment on which one would be most useful in providing the maximum out-of-band attenuation to a signal that is 75% of the center frequency

Answer 1

Chebyshev and Butterworth filters are designed for totally different applications. Butterworth filters are used in applications where maximum pass band flatness is required & Chebyshev filters are optimized to give a steeper roll off.

Percentage bandwidth is a simple calculation, and gives a normalized measure of how much frequency variation a system or component can handle. As you go higher in frequency, the absolute bandwidth as a part will naturally increase, while its percent bandwidth will decrease.

If you know the center frequency and the bandwidth, the percent bandwidth is:

BW%=BW/FC

Here "BW" is the absolute bandwidth and FC is the center frequency. Thus a filter with 1 GHz passband centered at 10 GHz will have 10% bandwidth, while a filter with 10 GHz bandwidth at 100 GHz will have the same 10%.

Alternatively, you can calculate BW% from two corner frequencies FL and FH:

BW%=2(FH/FL-1)/(FH/FL+1)

Where FH is the upper (higher) frequency and FL is the lower frequency of the passband. This is simple math, it is derived from the center frequency being the arithmetic average between the upper and lower corner frequencies:

FC=(FH+FL)/2

The out-of-band rejection of a bandpass filter is the attenuation applied to any of the signal outside of the chosen frequency band. This is sometimes incorrectly referred to as optical density, or isolation. Out-of-band rejection specifies a rejection value centered around ?c, which is normally in dB. It is defined in relation to a filter’s peak efficiency

The magnitude response |H(jw)| of the butterworth filter decreases with increase in frequency from 0 to infinity, while the magnitude response of the Chebyshev filter fluctuates or show ripples in the passband and stopband depending on the type of the filter. The width of the transition band is more in Butterworth filter compared to the Chebyshev filter.

Chebyshev filter will give the maximum out-of-band attenuation to a signal.