Consider a square wave, which through the Fourier Series Expansion is shown to be made up of an infinite number of odd harmonic frequencies. In the graphic above I show the summation of the first three components. If these components are all delayed the same amount, the waveform of interest is intact when these components are summed. However, significant group delay distortion will result if each frequency component gets delayed a different amount in time.
The following may help give additional intuitive insight for those with some RF or analog background. Consider an ideal lossless broadband delay line such as approximated by a length of coaxial cable , which can pass wideband signals without distortion. The transfer function of such a cable is shown in the graphic below, having a magnitude of 1 for all frequencies given it is lossless and a phase negatively increasing in direct linear proportion to frequency. The longer the cable, the steeper the slope of the phase, but in all cases "linear phase".
The simplest mathematical explanation is that the a phase that is linear with frequency and a constant delay are Fourier Transform pairs. This is the shift property of the Fourier Transform.
Just to add to what's already been said, you can see this intuitively by looking at the following sinusoid with monotonically increasing frequency. Shifting this signal to the right or left will change its phase.
But note also that the phase change will be larger for higher frequencies, and smaller for lower frequencies.
Or in other words, the phase increases linearly with frequency. Thus a constant time shift corresponds to a linear phase change in the frequency domain. This means that the input signal will be weighted and shifted intact as a whole by the group delay of the filter. And this will be the case if the underlying filter has linear phase or generalized linear phase.
Note that if the input signal is of broadband type; i. Then what's the effect of a filter with nonlinear phase or frequency dependent group delay on the input signal? A simple example would be a complicated input signal considered as a sum of multiple wavepackets at different center frequencies.
After the filtering, each packet with a particular center frequency will be shifted delayed differently due to frequency dependent group delay. And this will be resulting in a change in the time-order or space order of those wave packets, sometimes drastically, depending on how nonlinear the phase is, which is called as dispersion in communications terminalology. Not only the composite waveshape, but also some event orders may be lost.
This kind of dispersive channels have severe effects such as ISI inter symbol interference on transmitted data. This property of linear phase filters, therefore, is also known as waveform-preserving property, which is applicable to narrowband signals in particular. An example where waveform is important, other than ISI as mentioned above, is in processing of images, where the Fourier transform phase information is of paramount importance compared to magnitude of Fourier transform, for intelligibility of the image.
The same, however, cannot be said for perception of sound signals due to a different kind of sensitivity of the ear to the stimulus. The answer to this question is already been explained clearly in the previous replies. Yet I wish to give it a try to present a mathematical interpretation of the same. So if the phase is linear then all the frequency components of the signal will undergo the same amount of delay in time-domain which results in shape preservation.
This site uses cookies to deliver our services and to show you relevant ads and job listings. By using our site, you acknowledge that you have read and understand our Cookie Policy , Privacy Policy , and our Terms of Service. It is generally accepted that zero or linear phase filters are ideal for audio applications. This is because such filters delay all frequencies by the same amount, thereby maximally preserving waveshape.
Mathematically, all Fourier-components passed by the filter remain time-synchronized exactly as they were in the original signal. However, this section will argue that a phase response somewhere between linear- and minimum-phase may be even better in some cases. We show this by means of a Matlab experiment comparing minimum-phase and zero-phase impulse responses.
The matlab code is shown in Fig. An order elliptic-function lowpass filter [ 64 ] is designed with a cut-off frequency at 2 kHz. This is readily seen for an FIR filter,. Analysing Eqn. A digital filter is said to be bounded input, bounded output stable , or BIBO stable, if every bounded input gives rise to a bounded output.
Analyzing Eqn. Consequently, it is sufficient to say that a bounded input signal will always produce a bounded output signal if all the poles lie inside the unit circle. The zeros on the other hand, are not constrained by this requirement, and as a consequence may lie anywhere on z-plane, since they do not directly affect system stability. Therefore, a system stability analysis may be undertaken by firstly calculating the roots of the transfer function i.
Applying the developed logic to the poles of an IIR filter, we now arrive at a very important conclusion on why IIR filters cannot have linear phase. A BIBO stable filter must have its poles within the unit circle, and as such in order to get linear phase, an IIR would need conjugate reciprocal poles outside of the unit circle, making it BIBO unstable. However, a discussed below, phase equalisation filters can be used to linearise the passband phase response.
A second order Biquad all-pass filter is defined as:. Notice how the numerator and denominator coefficients are arranged as a mirror image pair of one another. The mirror image property is what gives the all-pass filter its desirable property, namely allowing the designer to alter the phase response while keeping the magnitude response constant or flat over the complete frequency spectrum. Cascading an APF all-pass filter equalisation cascade comprised of multiple APFs with an IIR filter, the basic idea is that we only need to linearise the phase response the passband region.
The other regions, such as the transition band and stopband may be ignored, as any non-linearities in these regions are of little interest to the overall filtering result. The APF cascade sounds like an ideal compromise for this challenge, but in truth a significant amount of time and very careful fine-tuning of the APF positions is required in order to achieve an acceptable result.
Nevertheless, despite these challenges, the APF equaliser is a good compromise for linearising an IIRs passband phase characteristics. ASN Filter Designer provides designers with a very simple to use graphical all-phase equaliser interface for linearising the passband phase of IIR filters.
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