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The essence and importance of linear phase property lies in the definition and the effect of group delay $$\tau(\omega) = - \frac {d\phi(\omega)}{d\omega}$$ on the applied signal $x[n]$, where $\phi(\omega)$ is the phase response of the filter; (phase of its frequency response).

Assume that a filter, with a fixed group delay of $n_0$ samples, is applied a narrowband input signal $x[n]$. Then the output signal will be (approximately) of the form $y[n] = K x[n-n_0]$ where $K$ is the filter gain evaluated at the center frequency of the narrowband input signal $x[n]$. This means that the input signal will be weighted and shifted intact as a whole by the group delay of the filter. And this can only happen when the group delay is independent of the frequency $\omega$. 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.e., its minimum and maximum frequencies are far from its center frequency, then the approximation is not valid and eventhough the group delay would still be the same for each sinusoidal component in the signal, their relative output amplitudes will differ by the frequency dependent filter gain $K(w)$.

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 essence and importance of linear phase property lies in the definition and the effect of group delay $$\tau(\omega) = - \frac {d\phi(\omega)}{d\omega}$$ on the applied signal $x[n]$, where $\phi(\omega)$ is the phase response of the filter; (phase of its frequency response).

Assume that a filter, with a fixed group delay of $n_0$ samples, is applied a narrowband input signal $x[n]$ (a signal having sufficiently narrow bandwidth across which any filter's arbitrary frequency-response magnitude remains approximately constant). Then the output signal will be (approximately) of the form $y[n] = K x[n-n_0]$ where $K$ is the filter gain evaluated at the center frequency of the narrowband input signal $x[n]$. If a filter has a flat passband of arbitrary large bandwidth, then the signal's bandwidth remaining inside the passband of filter is sufficient, and narrowband condition, of generality, is not necessary then. Eventually, for a linear phase filter, the input signal will be weighted and shifted intact as a whole by the group delay of the filter. Indeed, this can happen only when the filter's group delay is independent of the frequency $\omega$. And it's so if the underlying filter has linear phase (or generalized linear phase). Note that if the input signal is of broadband type, then the approximation of constant filter gain may not be valid in general, and eventhough filter may still be a linear-phase one, the output amplitudes will be weighted by frequency dependent gain $K(w)$ of filter, hence $y[n] \neq K x[n-n_0]$ anymore.

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 essence and importance of linear phase property lies in the definition and the effect of group delay $$\tau(\omega) = - \frac {d\phi(\omega)}{d\omega}$$ on the applied signal $x[n]$, where $\phi(\omega)$ is the phase response of the filter; (phase of its frequency response).

Assume that a filter, with a fixed group delay of $n_0$ samples, is applied a narrowband input signal $x[n]$. Then the output signal will be of the form $y[n] = K x[n-n_0]$ where $K$ is a frequency dependent filter gain evaluated at the center frequency of the narrowband input signal $x[n]$. This means that the input signal will be (delayed) shifted intact as a whole by the group delay of the filter. And this can only happen when the group delay is independent of the frequency $\omega$. And this will be the case if the underlying filter has linear phase (or generalized linear phase).

When this is not so then what happens? i.e., 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 frequencies. After the filtering, each packet with a particular 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. Not only the composite waveshape, but also some event orders may be lost.

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 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 essence and importance of linear phase property lies in the definition and the effect of group delay $$\tau(\omega) = - \frac {d\phi(\omega)}{d\omega}$$ on the applied signal $x[n]$, where $\phi(\omega)$ is the phase response of the filter; (phase of its frequency response).

Assume that a filter, with a fixed group delay of $n_0$ samples, is applied a narrowband input signal $x[n]$. Then the output signal will be (approximately) of the form $y[n] = K x[n-n_0]$ where $K$ is the filter gain evaluated at the center frequency of the narrowband input signal $x[n]$. This means that the input signal will be weighted and shifted intact as a whole by the group delay of the filter. And this can only happen when the group delay is independent of the frequency $\omega$. 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.e., its minimum and maximum frequencies are far from its center frequency, then the approximation is not valid and eventhough the group delay would still be the same for each sinusoidal component in the signal, their relative output amplitudes will differ by the frequency dependent filter gain $K(w)$.

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 essence and importance of linear phase property lies in the definition and the effect of group delay $$\tau(\omega) = - \frac {d\phi(\omega)}{d\omega}$$ on the applied signal $x[n]$, where $\phi(\omega)$ is the phase response of the filter; (phase of its frequency response).

Assume that a filter, with a fixed group delay of $n_0$ samples, is applied a narrowband input signal $x[n]$. Then the output signal will be of the form $y[n] = K x[n-n_0]$ where $K$ is a frequency dependent filter gain evaluated at the center frequency of the narrowband input signal $x[n]$. This means that the input signal will be (delayed) shifted intact as a whole by the group delay of the filter. And this can only happen when the group delay is independent of the frequency $\omega$. And this will be the case if the underlying filter has linear phase (or generalized linear phase).

When this is not so then what happens? i.e., 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 frequencies. After the filtering, each packet with a particular 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. Not only the composite waveshape, but also some event orders may be lost.

This property of linear phase filters, therefore, is also known as waveform-preserving property. An example where waveform is important 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. Also for industrial applications as well, the property becomes important when the underlying process is sensitive to wave-shape.

The essence and importance of linear phase property lies in the definition and the effect of group delay $$\tau(\omega) = - \frac {d\phi(\omega)}{d\omega}$$ on the applied signal $x[n]$, where $\phi(\omega)$ is the phase response of the filter; (phase of its frequency response).

Assume that a filter, with a fixed group delay of $n_0$ samples, is applied a narrowband input signal $x[n]$. Then the output signal will be of the form $y[n] = K x[n-n_0]$ where $K$ is a frequency dependent filter gain evaluated at the center frequency of the narrowband input signal $x[n]$. This means that the input signal will be (delayed) shifted intact as a whole by the group delay of the filter. And this can only happen when the group delay is independent of the frequency $\omega$. And this will be the case if the underlying filter has linear phase (or generalized linear phase).

When this is not so then what happens? i.e., 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 frequencies. After the filtering, each packet with a particular 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. Not only the composite waveshape, but also some event orders may be lost.

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 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.