Signal processing: basics, techniques and applications

Before digital signal processing, continuous analog signals are converted into discrete digital signals by an analog-to-digital converter (ADC). These digital signals can then be processed with digital signal processors or computers. Digital signal processing offers several advantages: 

• Reproducible and precise results: Environmental influences and component tolerances have little impact on data processing

   

• Flexible possibilities: Fast development and inexpensive modification of signal processing

   

• Simple implementation through the use of known algorithms

   

• Secure storage and efficient transmission of digital data

   

• Improved signal-to-noise ratio through adaptation to the data

   

Convolution and digital signal processing filters
 

Convolution is a mathematical operator that combines two functions to produce a third function. In digital signal processing, convolution is often used to perform filter operations. Filters can take various forms such as low-pass, high-pass, band-pass or notch filters. They are used to remove unwanted frequency components from a signal or to emphasize certain frequency ranges. This improves the data during signal processing, for example by smoothing or noise suppression. 

Use of the Fourier transform
 

An essential part of signal processing is the Fourier transform, which transforms a signal from its time domain into the frequency domain. This technique is of central importance because many signals can be analyzed and manipulated more easily in the frequency domain. By using the Fast Fourier Transform (FFT), an efficient implementation of the Fourier transform, engineers can quickly identify and analyze the frequency components of a signal. 

Mathematical representation

Difference equations are usually represented in the form y(n) = ∑_k=0^N a_k x[n − k] + ∑_m=1^M b_m y[n − m] where y(n) the current output value, x(n) the current input value, a_k and b_m are the filter coefficients. This equation shows how the current output value is determined by a weighted sum of the current and previous input values and the previous output values. By selecting suitable coefficients, different filter characteristics such as low-pass, high-pass, band-pass or band-stop filters can be realized.

Application in IIR and FIR filters

In practice, difference equations enable the implementation of Infinite Impulse Response (IIR) and Finite Impulse Response (FIR) filters. IIR filters, which are described by recursive difference equations, use both past input and output values and can achieve complex filter properties with less computational effort. FIR filters, on the other hand, only use past input values and are always stable, but are often more complex to calculate.