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## Frequency Transformations

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In this chapter we discuss the realization of low-pass (LP), high-pass (HP), bandpass (BP), and bandstop (BS) filters by applying suitable frequency transformations to the normalized LP filter. These transformations generally preserve the LP magnitude response (attenuation); other LP characteristics are often retained, however, especially in the case of the narrow-band BP filter. We also give the necessary equations (where possible) for denormalizing the LP responses, allowing them to characterize the appropriate filter type. When the transformation preserves the important attributes of the LP response, the tedious and time consuming approximation step in the design sequence is eliminated. This is the big advantage of the frequency-transformation approach. Furthermore, if the transformation function is of the same form as a reactance function, the filter element values are easily determined. Unfortunately certain LP characteristics are distorted by the transformation, and to avoid these distortions we either re-solve the approximation problem or use a different transformation. We present an example of each approach. The first results in the wide-band constant-delay filter and the second results in the low -transient HP filter. This chapter also includes narrow-band BP filter design, which is applicable not only to lumped-constant filters but also to transmission line and waveguide filters.

Chapter Contents:

• 4.1 Normalized Parameters
• 4.2 Low-Pass Filter
• 4.2.1 Transient Responses
• 4.2.2 Element Values
• 4.2.3 Example of Low-Pass Calculation
• 4.3 High-Pass Filter
• 4.3.1 Conventional Transformation
• 4.3.2 Transient Responses
• 4.3.3 Relationship Between LP and HP Transient Responses
• 4.3.4 Element Values
• 4.3.5 Example of High-Pass Calculation
• 4.3.6 Preservation of LP Transient Characteristics
• 4.3.7 Application of New High-Pass Transformation
• 4.4 Bandpass Filter
• 4.4.1 Conventional Transformation
• 4.4.2 Group Delay Behavior
• 4.4.3 Pole-Zero Locations
• 4.4.4 Element Values
• 4.4.5 Example of Bandpass Calculation
• 4.5 Narrow-Band Bandpass Filter
• 4.5.1 Basic Definitions
• 4.5.2 Example of Attenuation Calculation
• 4.5.3 Pole-Zero Locations
• 4.5.4 Group Delay Behavior
• 4.5.5 Example of Group Delay Calculation
• 4.5.6 Example of Wide-Band Group Delay Calculation
• 4.5.7 Transient Responses
• 4.5.8 Realizations with Nodal and Mesh Networks
• 4.5.9 Example of Network Design
• 4.5.10 Features of Nodal and Mesh Networks
• 4.6 Wide-Band Constant-Delay Bandpass Filter
• 4.6.1 Transfer Function and Least-Squares Approximation
• 4.6.2 Tables of Optimum Parameters
• 4.6.3 Normalized Attenuation and Group Delay Curves
• 4.6.4 Element Values
• 4.7 Bandstop Filter
• 4.7.1 Conventional Transformation
• 4.7.2 Example of Bandstop Calculation
• 4.7.3 Narrow-Band Transformation
• References
• Problems

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