Compare the outputs. Keep in mind that both of the signals contain same frequencies. Please refer some good books. This property can be important in communications applications where you might want to filter out noise but not affect the shape of a received pulse.
Or in audio applications, if you want to filter out high frequencies without affecting the phase of the signal components in the pass band. A linear phase FIR filter is commonly used in audio sample rate conversion as well as analog-to-digital and digital-to-analog converters that use over-sampling.
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Registration is free. Click here to register now. 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.
As seen below, the interface is very intuitive, and allows designers to quickly place and fine-tune APF filters positions with the mouse. Right clicking on the frequency response chart or on an existing all-pass design marker displays an options menu, as shown on the left.
Designing an equaliser composed of three APF pairs, and cascading it with the Chebyshev filter of Figure 1, we obtain a filter waveform that has a much a sharper peak with less attenuation and oscillation than the original IIR — see below.
However, this improvement comes at the expense of three extra Biquad filters the APF cascade and an increased group delay, which has now risen to 24 samples compared with the original 10 samples. The frequency response of both the original IIR and the equalised IIR are shown below, where the group delay shown in purple is the average delay of the filter and is a simpler way of assessing linearity.
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