Network Analysis and Synthesis

1. Network Analysis and Synthesis Chapter 7 Filter types, specifications

2. 7.1 Introduction • Filters can be considered as frequency selective networks. • A filter is required to separate unwanted signal from a mixture of unwanted and wanted signals. • The transfer characteristic of a filter is so shaped that the ratio of unwanted to wanted signal at the output of the filter is minimized.

3. Filter transfer characteristics are usually given interims of filter specifications. • The filter specifications are then given in terms of cut-off frequencies (pass band frequencies) and stop-band frequency. • Pass-band:-the frequency band of wanted signals. • Stop-band:- the frequency band of unwanted signals. • Cut-off frequency:- The frequency associated with the boundary between stop band and an adjacent pass-band. It is the frequency at which the output is 0.707 times the maximum value in the pass band.

4. An ideal filter should pass the wanted signals with no attenuation and provide infinite attenuation for the unwanted signals. • Electric filters are ubiquitous and it is difficult to conceive a modern electronic device or system which does not employ and electric filter. 7.2 Types of filters • According to their pass band and stop band frequencies, filters are categorized as: • Low pass filters • Band pass filters • High pass filters • Band reject filters

5. 7.2.1 Low pass filters • A low-pass filter is a filter that passes low-frequency signals but attenuates (reduces the amplitude of) signals with frequencies higher than the cutoff frequency. • The magnitude and phase of an ideal low pass filter is • wp is the pass band frequency.

6. The impulse response of the ideal filter is • Note that this impulse response is not causal, hence, not realizable. • In general, a band limited frequency response leads to non-causal impulse response.

7. In order to circumvent this limitation, filter specifications are given as follows

8. In words: • The attenuation should be less than αp in the pass band. • The attenuation should be greater than αs in the stop band. • The attenuation can be anything in the transition band (the frequency band between the pass band and the stop band).

9. 7.2.2 Band pass filters • A band-pass filter is a device that passes frequencies within a certain range and rejects (attenuates) frequencies outside that range. • It passes signals between two cutoff frequencies but attenuates(reduces the amplitude of) the rest.

10. Again this ideal band-pass filter is not realizable. Hence, a realizable band-pass filter specification is:

11. For a geometrically symmetric band-pass filter, the following equation is satisfied. • Hence,

12. 7.2.3 High pass filters • A high-pass filter is a filter that passes high-frequency signals but attenuates signals with frequencies lower than the cutoff frequency. • An ideal high-pass filter is • This is not realizable frequency response however.

13. A realizable high pass filter specification is ωs ωp

14. 7.2.4 Band reject filters • Band-reject filters are the exact opposite of band pass filters. • They attenuate signals between two frequency ranges but pass the rest. • The can be obtained by the following equation

15. 7.3 Frequency transformation • Most of the widely used filter synthesis methods (will be discussed in next chapter) are for low pass filters only. • So how do we synthesize band-pass, high-pass and band-reject filters? We use frequency transformation methods.

16. By frequency transformation we mean: • Transform the filter specifications to normalized low-pass filter specification. • Obtain the normalized low-pass filter transfer function using modern filter synthesis methods. • Transform the low-pass transfer function back to the required type of filter. • Synthesize the filter. • Steps 3 and 4 can be exchanged.

17. 7.3.1 Band-pass to Low-pass • The transformation from band-pass to low pass is

18. Note that • This means that the frequency range [wp2,wp1] has been transformed to [-1,1], hence our new specification will have a low pass form.

19. Procedure for designing band pass filter • Transform the given specifications to normalized low pass specifications with • Design the normalized low pass filter with the appropriate method • Transform the obtained normalized low pass filter transfer function back to band pass filter with

20. 7.3.2 High pass to low pass • The transformation from high pass to low pass is • Note that:

21. Procedure for designing high pass filter • Transform the given specifications to normalized low pass specifications with • Design the normalized low pass filter with the appropriate method • Transform the obtained normalized low pass filter transfer function back to high pass filter with

22. 7.3.3 Band-reject to low pass

23. Procedure for designing band pass filter • Transform the given specifications to normalized low pass specifications with • Design the normalized low pass filter with the appropriate method • Transform the obtained normalized low pass filter transfer function back to band pass filter with

24. If we interchange steps 3 and 4 • We can use the following element substitutions