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Linear Phase Filters

Key Takeaways

  • Linear phase filters maintain a uniform group delay across all frequencies, ensuring signal integrity by preserving the waveform shape.

  • FIR (Finite Impulse Response), Chebyshev Type II, Bessel Filter, and Raised Cosine each have unique characteristics and applications.

  • The linear phase is critical in maintaining the waveform's integrity in various applications. For instance, in digital communications, it prevents signal spreading and distortion, while in audio processing, it ensures time-aligned frequencies for accurate sound reproduction.

Filter A has a linear phase response, whereas Filter B exhibits a non-linear phase

Filter A has a linear phase response, whereas Filter B exhibits a non-linear phase. By examining the frequency response, we can see that a filter possesses a linear phase when its phase response displays a consistently uniform slope. Looking at the output, we can see the waveform shape is maintained in filter A output, but not in filter B output.

Linear phase filters are filters that have a phase response that linearly correlates with frequency. This means every frequency component of an input signal experiences an identical time shift, commonly known as group delay. This uniformity in phase shift across all frequencies means that each frequency component of the signal is delayed equally (called group delay), preserving the signal's integrity over time. Group delay is crucial for minimizing signal distortion and spread over time, making these filters ideal for applications where signal fidelity is paramount.

Linear Phase Filter Types

Filter Type



Common Applications

FIR (Finite Impulse Response)

Designed using various windowing techniques

  • Inherently linear phase

  • Symmetric or antisymmetric impulse response

  • Signal processing

  • Audio processing

Chebyshev Type II

Inverse Chebyshev filter

  • Flat passband and equiripple stopband

  • Linear phase in the passband

  • Image processing

  • Communications systems

Bessel Filter

Designed for optimal pulse preservation

  • Maximally flat group delay

  • Gentle roll-off in frequency response

  • Audio processing

  • Data transmission

Raised Cosine

Used to shape digital signals for efficient transmission

  • Linear phase characteristic

  • Prevents intersymbol interference

  • Digital communication systems

Linear Phase Filter Basics

It's crucial to understand that filters impact both the amplitude and phase of a signal. For instance, a basic resistor-capacitor (RC) low-pass filter can shift the phase of an output sinusoid up to 90 degrees compared to the input, but this depends on the signal's frequency.

To ensure equal time delays for all frequencies, each frequency needs a specific phase shift that results in the same delay across the board. As frequency rises, a fixed phase shift translates into a shorter time span, necessitating a greater phase shift for compensation. The group delay, being proportional to the derivative of the phase response with respect to frequency, is constant in such a filter because the derivative of a linear function is a constant. This is why linear phase response is synonymous with constant group delay. The Bessel filter is a well-known example optimized for linear phase characteristics.

Discrete Time Signals

For discrete-time signals, attaining a perfect linear phase is straightforward with finite impulse response (FIR) filters. This is accomplished by designing the filter coefficients to be symmetric or anti-symmetric. While FIR filters can achieve perfect linear phase, infinite impulse response (IIR) filters offer an approximation of linear phase. IIR filters are generally more computationally efficient than FIR filters, and for instances where an approximate linear phase is necessary, they are more common.

Finite Impulse Response Filters

Finite impulse response (FIR) filters have exactly linear phases. There are two main methods for designing linear phase FIR filters: 

  1. Window-based design method 

  2. The Remez design method

The Remez design method, particularly as implemented in the DFD Remez Design VI, stands out for its superior power and flexibility compared to window-based design methods. 

FIR Filter Classification

This table categorizes the frequency characteristics of digital filters, specifically in the context of Finite Impulse Response (FIR) filters. Each type (I, II, III, IV) describes a specific symmetry and order characteristic of the filter's frequency response, A(f). 



Frequency Characteristics


Common Applications


Even-order, Symmetric

  • A(f) is symmetric about f = 0 and f = 0.5

  • A(f) is periodic with period 1

General-purpose filtering, non-zero at all frequencies

General purpose filtering


Odd-order, Symmetric

  • A(f) is symmetric about f = 0 and antisymmetric about f = 0.5

  • A(f) is constrained to 0 at f = 0.5

  • A(f) is periodic with period 2

Smoothing applications, zero response at Nyquist frequency

Smoothing applications


Even-order, Antisymmetric

  • A(f) is antisymmetric about f = 0 and f = 0.5

  • A(f) is constrained to 0 at both f = 0 and f = 0.5

  • A(f) is periodic with period 1

Differentiators or Hilbert transformers, linear phase, no DC component

Differentiators and

Hilbert transofrmers


Odd-order, Antisymmetric

  • A(f) is antisymmetric about f = 0 and symmetric about f = 0.5

  • A(f) is constrained to 0 at f = 0

  • A(f) is periodic with period 2

Specific phase response requirements, similar to Type III but with different filtering characteristics

Phase shifters

It’s important to note that not all FIR filters automatically have a linear phase. This characteristic is contingent on the symmetry or anti-symmetry of the filter's coefficients or taps around a central point. In practical terms, this means the first and last coefficients are identical, the second and second-to-last are the same, and so forth. 

The formula to determine the delay introduced by a linear-phase FIR filter is straightforward. For an FIR filter with N taps, the delay is calculated as (N – 1) / (2 * Fs), where Fs represents the sampling frequency. For example, a 21-tap linear-phase FIR filter operating at a 1 kHz sampling rate would have a delay of (21 – 1) / (2 * 1 kHz) = 10 milliseconds.

Why the Linear Phase Is Important

Preserving wave shapes is crucial in many applications.

  • In complex modulation schemes such as 128 QAM, decisions are made based on the waveshape, often in a quadrature space involving multiple thresholds. Accurately discerning whether a received signal represents a "1" or "0" depends on maintaining the integrity of the originally transmitted waveshape. Any distortion could lead to incorrect thresholding decisions, resulting in bit errors within the communication system.
  • In radar signal processing, the waveshape of the returned signal is vital. It can carry significant information about the properties of the target, such as size, shape, or even material composition. Preserving the original waveshape of these signals is key to extracting accurate and detailed information about the target.
  • In audio processing, the debate continues over the importance of time aligning different components of a complex waveshape. Some experts argue this alignment is essential for preserving or enhancing subtle qualities of the listening experience, such as the stereo image. Furthermore, in audio, it's important for frequencies representing different pitches to remain in sync to ensure accurate sound reproduction.
  • In digital communications, the harmonics that make up a square wave must have a consistent delay to prevent distortion.

What Happens if the Filter Has a Nonlinear Phase Response?

In digital communications, phase distortion can lead to signal spreading. This phenomenon can cause interference between closely timed information symbols, impacting the integrity and clarity of the communication. When different groups of frequencies undergo varying delays, a phenomenon referred to as group-delay error, it can alter or color the sound, producing audible artifacts.

Cadence’s AWR Supports Design and Simulation

Understanding the nuances of linear phase filters is crucial for optimal filter design and simulation. Explore how Cadence AWR's advanced tools can help you efficiently integrate these principles into your RF and microwave designs for superior performance and accuracy. 

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