Signal processing
Another branch of applied mathematics is signal processing, which finds its application in the engineering field, focusing on analyzing and processing signals such as sound, images, scientific measurements, and filtering out noise. Signal processing deals with the transformation of a signal from time-series to hyper-spectral images, which are obtained from different electromagnetic measurements. Classic transformations of signals such as spectrograms and wavelets are often used with ML techniques. Such representations can also be used as inputs to deep neural networks. The Kalman filter is one classic signal processing filter that uses a series of measurements over time to produce estimates of unknown variables.
Understanding the problem
A signal is a function of a continuous variable, such as time or space. An analog signal is transformed into a digital signal by sampling it at specified intervals of time called the sampling period, the inverse of which is the sampling rate (per second or Hertz). The sampling rate has to be at least twice as high as the maximum frequency of the analog signal. It establishes a sufficient condition that permits a discrete sequence of samples to encapsulate all the information from a continuous time signal into a discrete time signal.
Figure 1.3: 60 kHz sinusoidal (Hann-windowed) tone burst in the time domain and frequency domain of the signal
The frequency domain representation of a signal is done with the Discrete Fourier Transform (DFT). The Fast Fourier Transform (FFT) is an efficient computation method of DFT. FFT is rarely applied over the entire signal (speech signal, for example) at once but rather in frames due to the stochastic nature of the signal, an example of which is illustrated in Figure 1.3. FFT is available as a library function with scipy for the computation of the frequencies of each frame. A type of Fourier transform called the Short-time Fourier Transform (STFT) is typically applied on each individual frame.
Formulation of the problem
It is clear that Discrete-Time Signal Processing (DSP) is meant for sampled signals and establishes a mathematical basis for DSP, which is essentially analyzing and modifying a signal to improve (or optimize) its efficiency or performance. By using DFT, a discrete sequence can be represented as its equivalent frequency ‘
domain. The linearity property of the Fourier transform yields two signals, 
and 
:
Where 
and 
are the Fourier transforms of 
and 
respectively, a concept often used in the filtering of signals, which is the transformation of the time 
domain to the frequency 
domain. The duality property of the Fourier transform is useful as it enables solving complex ones that otherwise would be difficult to compute directly. It yields that if x
has a Fourier transform 
, then one can form a new function of time 
that has a functional form of the transformation, for example:
A time shift affects the frequency, and a frequency shift affects the time of the functions. Let us take an example of a spectrogram to understand DSP.
A spectrogram displays the spectrum of frequencies of a waveform over time and is extensively used in the fields of music and speech processing and radars. It is generated by an optical spectrometer, a Fourier transform, or a wavelet transform and is usually depicted as a heat map wherein the strength or intensity of the signal changes with the color (brightness). To generate a spectrogram, a time-domain signal is divided into chunks of equal lengths that usually overlap, and FFT is applied to each chunk for the calculation of the frequency range. The spectrogram is a plot or graph of the spectrum on each segment or FFT frame, as a frequency versus a time image (or a 3D surface), shown in Figure 1.4, and the third dimension (represented by the color bar) indicates the amplitude of a particular frequency at a particular time. This process corresponds to the computation of the squared magnitude of STFT of the signal.
Figure 1.4: Spectrogram
Spectrograms can be used to identify characteristics of non-stationary or non-linear signals as a collection of time-frequency analyses. The parameters in a spectrogram typically are frame count (number of FFTs making it up), frequency range (minimum and maximum), FFT spacing, and FFT width (width of time each FFT represents).
Spectrograms are used with recurrent neural networks (RNNs) in speech recognition, as a primary example. We learned about how digital signals are free (well, almost) of noise and less distorted in this sub-section, and in the next, we are going to explore control theory, another mathematical modeling technique widely used in industrial processes. Control theory is, in general, useful whenever feedback happens in either regulator or servo mechanisms, for example, navigation systems and industrial production processes.