[Reprinted] response frequency and group latency

Although the phase frequency response should reflect the processing time of different frequency signals, it does not mean that the larger the phase frequency response, the longer the system processing time. We can know from a simple sine signal exp (J * w * n) that the phase is w * n, that is, the phase is not only related to time, but also frequency. In signal processing, group delay is another concept used to characterize the system delay time. Its mathematical definition is as follows:

The preceding definition clearly shows that group latency is more specific to the time delay of the system for signals of different frequencies. So we naturally want to ask, what is the difference between group latency and phase frequency response?

Let's look at a specific example. Assume that a signal is composed of two sine signals with different frequencies, as shown in waveform 1 (. If the signal is passed through a inverter, that is, the system that times the signal by-1, it is clear that the phase frequency of the inverter should be Fai = pi. The signal waveform 1 (B) after this system is used is shown. If this signal is transmitted through a distortion-free transmission system, that is, the unit impulse response of the system is H (n) = delta (n-n0), where N0 is a constant. Apparently, the group latency of a non-distortion system is Tao = N0. The signal waveform after the system is used is shown in 1 (c. Comparing the waveforms of these three signals, we can see that the signal (B) obtained after the inverter is different from the original signal (A) in details. The signal (c) after a system without distortion is exactly the same as the original signal in detail, but it only delays time.

From the above example, it can be seen that the phase frequency response and group latency both reflect the latency of the system for different frequency signals, but their meanings are still different. The phase frequency response should reflect the relative value of the system for the input signal delay, and the group delay reflects the absolute value of the system for the input signal delay. For signals with complex frequency components, the phase frequency response should be a constant, which will lead to distortion of the signal. A system with group delay as a constant will not produce distortion of the signal. The requirement that signals pass through the system without distortion is an internal requirement in many application scenarios. For example, the transmission of communication signals requires the group delay of the system to be a constant. The formula for calculating group latency shows that if group latency is a constant, the corresponding phase frequency response should be in the form of FAI =-W * N0, which is also called the linear phase of the system.

In actual signal processing, group latency is often used to determine whether the system produces distortion on the input signal. Therefore, in some cases, it is also called envelope latency. The use of phase frequency response is much more extensive, because on the one hand, the calculation of group latency requires differentiation, and the calculation of Phase Frequency Response after fuzzy decoding is also required, which is complicated; on the other hand, the physical meaning of group latency can also be well reflected in the phase frequency response. For example, the linear phase completely shows the group latency as a constant, this also shows that the phase frequency response should be a broader concept than the group delay.

Source Address: http://blog.csdn.net/deepdsp/article/details/7230815

[Reprinted] response frequency and group latency