Signal Processing - Time-Frequency Analysis
foreword
Almost all signals in nature are non-stationary signals. For example, our speech signal is a typical non-stationary signal. So what is a stationary signal and a non-stationary signal? A common understanding is that the statistical characteristics (such as expectation, variance, etc.) of the sampled values obtained by the stationary signal at different times are the same, while the non-stationary signal is the opposite, and its characteristics will change with time. In signal processing, this characteristic is usually referred to as frequency.
Usually the Fourier transform is only suitable for dealing with stationary signals. For non-stationary signals, since the frequency characteristic will change with time, in order to capture this time-varying characteristic, we need to perform time-frequency analysis on the signal, including short-time Fourier transform, wavelet Transform, Hilbert transform, and Hilbert-Huang transform. The analysis is introduced one by one below.
1. Fourier Transform (FFT)
First consider the Fourier transform and its inverse transform of a continuous signal, as follows:
In practical applications, computers can only process discrete signals, so the continuous signal x(t) is sampled in the time domain to obtain a set of discrete samples x( n), perform Fourier transform on it to get:
the above formula is the discrete time Fourier transform (DTFT), since the frequency domain value obtained after the transformation is still continuous, we continue to sample the frequency domain to get:
above The formula is the discrete Fourier transform (DFT), and the fast Fourier transform (FFT) commonly used in current computers is a fast algorithm of DFT.
However, the Fourier transform is a global transformation. After the time-domain signal undergoes Fourier transform, it becomes a frequency-domain signal. The time-domain information cannot be seen from the frequency domain. We can learn from the previous section Fourier transform and inverse transform formulas are explained. When performing forward transform, the integration interval is the entire time domain, so the transform result will not contain time domain information. The same is true for inverse transform. It is difficult for us to analyze the characteristics of non-stationary signals only through time-domain signals or amplitude spectra. Next we will introduce a time-frequency analysis (Time-Frequency Analysis) method - short-time Fourier transform (STFT).
2. Short-time Fourier transform (STFT)
The short-time Fourier transform is defined as:
where x(m) is the input signal and w(m) is the window function, which is reversed in time and has an offset of n samples. X(n,w) is a two-dimensional function of time n and frequency f, which connects the time domain and frequency domain of the signal, and we can perform time-frequency analysis on the signal.
Two kinds of spectrograms can be obtained by using different window lengths when calculating the spectrogram, that is, narrow-band and wide-band spectrograms. Long time windows (at least two pitch periods) are often used to compute narrowband spectrograms, while short windows are used to compute wideband spectrograms. The narrow-band spectrogram has higher frequency resolution and lower time resolution. Good frequency resolution can make each harmonic component of speech easier to distinguish, which is displayed as horizontal stripes on the spectrogram. On the contrary, broadband spectrogram has higher time resolution and lower frequency resolution, low frequency resolution can only get spectral envelope, and good time resolution is suitable for analyzing and testing the pronunciation of English speech.
It can be seen that for the short-time Fourier transform with a fixed frame length, the time resolution and frequency resolution in the global range are fixed. If we want to have high frequency resolution in the low frequency region and high time resolution in the high frequency region, obviously STFT cannot meet the requirements. We continue to introduce another time-frequency analysis method - wavelet transform.
3. Wavelet Transform (WT)
For any energy-limited signal f(t), its continuous wavelet transform (CWT) is defined as:
wavelet transform has time-frequency windows of different sizes at different times and frequencies, and can achieve higher frequency resolution in the low-frequency region, but its Still limited by the Heisenberg uncertainty principle, time resolution and frequency resolution cannot have the best of both worlds. At the same time, the time-frequency window of wavelet transform is not completely self-adaptive, it also needs to choose the basis function artificially.
The above methods are all limited by Heisenberg's uncertainty principle, and they are not fully adaptive methods. Next, a time-frequency analysis method that is not limited by the Heisenberg uncertainty principle and has better adaptability - Hilbert Huang transform
4. Hilbert transform
Before introducing the Hilbert-Huang transform, we first introduce the Hilbert transform.
The Hilbert transform is also an extension of the Fourier transform. It is often used for modulation and demodulation in communication systems. Of course, it can also be used for time-frequency analysis of signals. Its calculation method is:
- Compute the FFT of the input signal, saved as a vector F
- Create a vector h where
- Calculate the inner product of F and h
- Calculate the iFFT of the sequence obtained in the previous step
In the field of time-frequency analysis, the Hilbert transform is mainly used for instantaneous frequency estimation. From the above analysis, it can be seen that the analytical signal of the original signal can be obtained by using the Hilbert transform, assuming that the analytical signal is z(t),
But we can't directly perform Hilbert transform on the multi-frequency component signal, we need to further process it to decompose the original signal into a superposition of single-frequency signals, which requires the EMD decomposition in the Hilbert-Huang transform.
5. Hilbert-Huang Transform
Compared with HT, HHT has one more empirical mode decomposition (Empirical Mode Decomposition, EMD). EMD is to decompose complex signals into several intrinsic mode functions (Intrinsic Mode Function, IMF) from high frequency to low frequency. IMF Two conditions need to be met:
- The number of signal extreme points is equal to the number of zero points or the difference is 1
- The local mean of the upper envelope defined by the maximum value and the lower envelope defined by the minimum value of the signal is 0 (that is, the envelope is symmetrical up and down)
A simple understanding is that EMD is the process of sequentially extracting the highest frequency component of each part of the signal, so each IMF is actually a single-frequency component signal, so that we can perform Hilbert transform on each IMF component to obtain Hilbert spectrum for each component.