1.1 Basic concepts of digital filtering
Digital filtering has specific characteristics that you need to pay special attention to. The analog input signal must satisfy certain requirements. Furthermore, on converting an output digital signal into analog form, it is necessary to perform additional signal processing in order to obtain the appropriate result.
Figure 1-1 shows the block diagram of digital filtering process.
Figure 1-1. Digital filtering
The process of converting an analog signal into digital form is performed by sampling with a finite sampling frequency fs. If an input signal contains frequency components higher than half the sampling frequency (fs/2), it will cause distortion to the original spectrum. This is the reason why it is first necessary to perform filtering of an input signal using a low-pass filter that eliminates high-frequency components from input frequency spectrum. This filter is called anti-aliasing filter as it prevents aliasing.
After the process of filtering and sampling, a digital signal is ready for further processing which, in this case, is filtering using the appropriate digital filter. The output signal is also a digital signal which, in some cases, needs to be converted back into analog form. After digital-to-analog conversion, signal contains some frequency components higher than fs/2 that must be eliminated. Again, it is necessary to use a low-pass filter with the sampling frequency fs/2. The specific characteristics of conversion affecting the signal are beyond the scope of this book.
Digital filter attenuation is usually expressed in terms of the logarithmic decibel scale (dB). The attenuation measured in decibels can be found using the following expression:
a = 20 * log(H(f))
Cut-off frequencies are used for filter specification, which will be discussed later. The cut-off frequency of the passband is a frequency at which the transition of the passband to the transition region occurs. The cut-off frequency of the stopband is a frequency at which the transition of the transition region to the stopband occurs. These two frequencies are equivalent only for the ideal filter which is not possible to realize in practice. In other words, they are always different.
