Time-Frequency Analysis

Abstract

Fourier-analysis provides a description of a given data set in terms of monochromatic oscillations without any time information. It is thus mostly useful for stationary signals. If the spectrum changes in time it is desirable to obtain information about the time at which certain frequencies appear. This can be achieved by applying Fourier analysis to a short slice of the data (short time Fourier analysis) which is shifted along the time axis. The frequency resolution is the same for all frequencies and therefore it can be difficult to find a compromise between time and frequency resolution. Wavelet analysis uses a frequency dependent window and keeps the relative frequency resolution constant. This is achieved by scaling and shifting a prototype wavelet - the so called mother wavelet. Depending on the application wavelets can be more general and need not be sinusoidal or even continuous functions. Multiresolution analysis provides orthonormal wavelet bases which simplify the wavelet analysis. The fast wavelet transformation connects a set of sampled data with its wavelet coefficients and is very useful for processing audio and image data.

Problems

Problem 8.1 Short Time Fourier Transformation

In this computer experiment STFT analysis of a frequency modulated signal

$$\begin{aligned} f(t)=\sin \varPhi (t)=\sin \left( \omega _{0}t+\frac{a\omega _{0}}{\omega _{1}}(1-\cos \omega _{1}t)\right) \end{aligned}$$

(8.198)

with a momentaneous frequency of

$$\begin{aligned} \omega (t)=\frac{\partial \varPhi }{\partial t}=\omega _{0}(1+a\sin \omega _{1}t) \end{aligned}$$

(8.199)

is performed and shown as a spectrogram (Figs. 8.10, 8.11). Sampling frequency is 44100 Hz, number of samples 512.

You can vary the carrier frequency $\omega _{0}$, modulation frequency $\omega _{1}$ and depth a as well as the distance between the windows. Study time and frequency resolution

Problem 8.2 Wavelet Analysis of a Nonstationary Signal

In this computer experiment, a complex signal is analyzed with Morlet wavelets over 6 octaves (Fig. 8.14). The signal is sampled with a rate of 44 kHz. The parameter d of the mother wavelet (8.61) determines frequency and time resolution. The frequency $\omega _{0}$ of the mother wavelet is taken as the Nyquist frequency which is half the sampling rate. The convolution with the daughter wavelets (8.76) is calculated at 400 times with a step size of 0.726 ms (corresponding to 32 samples)

$$\begin{aligned} t_{n}=t_{0}+n\varDelta t \end{aligned}$$

(8.200)

and for 300 different values of the scaling parameter

$$\begin{aligned} s_{m}=1.015^{m}. \end{aligned}$$

(8.201)

The signal consists of two sweeps with linearly increasing frequency of the form

$$\begin{aligned} f_{1,2}(t)=\sin \left[ \omega _{1,2}t+\frac{\alpha _{1,2}}{2}t^{2}\right] \end{aligned}$$

(8.202)

and another component which switches between a 5 kHz oscillation and the sum of a 300 Hz and a 20 kHz oscillation at a rate of 20 Hz

$$\begin{aligned} f_{3}(t)=\left\{ \begin{array}{c} \sin (\omega _{20kHz}t)+\sin (\omega _{300Hz}t) \quad \text { if }\sin (\omega _{20Hz}t)<0\\ \sin (\omega _{5kHz}t) \quad else. \end{array}\right. \end{aligned}$$

(8.203)

Study time and frequency resolution as a function of d

Problem 8.3 Discrete Wavelet Transformation

In this computer experiment the discrete wavelet transformation is applied to a complex audio signal. You can switch on and off different components like sweeps, dial tones and noise. The wavelet coefficients and the reconstructed signals are shown. (see Figs. 8.23, 8.24).

Fig. 8.23

(Wavelet coefficients of a complex audio signal) From Top to Bottom The black curve shows the input signal. The finest details in light green, red and blue correspond to a high frequency sweep from 5000–15000 Hz starting at 0.7 s plus some time dependent noise. Cyan, orange and maroon represent a sequence of dial tones around 1000 Hz, dark green and magenta show the signature of several rectangular 100 Hz bursts with many harmonics. The black curve at the Bottom shows the coefficients of the coarse approximation, which essentially describes random low frequency fluctuations. The curves are vertically shifted relative to each other

Fig. 8.24

(Wavelet reconstruction) The different contributions to the signal are reconstructed from the wavelet coefficients. Color code as in Fig. 8.23. The original signal (Top black curve) is exactly the sum of the coarse approximation (Bottom black curve) and all details (colored curves). The curves are vertically shifted relative to each other