Introduction
The purpose of this book is to provide you with different theorethical and practical approaches to digital filter design. The book covers design of both finite and infinite impulse response filters. In addition, it applies the most popular and most commonly used design methodes giving the best solutions. Intuitive explanations and numerous examples contained herein will help you to devolope your understanding of these methods and test your knowledge on concrete issues.
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.
1.2.Types of digital filters
Filters can be classified in several different groups, depending on what criteria are used for classification. The two major types of digital filters are finite impulse response digital filters (FIR filters) and infinite impulse response digital filters (IIR).
Both types have some advantages and disadvantages that should be carefully considered when designing a filter. Besides, it is necessary to take into account all fundamental characteristics of a signal to be filtered as these are very important when deciding which filter to use. In most cases, it is only one characteristic that really matters and it is whether it is necessary that filter has linear phase characteristic or not.
Speech signal, for example, can be processed in the systems with non-linear phase characteristic. The phase characteristic of a speech signal is not of the essence and as such can be neglected, which results in the possibility to use much wider range of systems for its prosessing.
There are also signals for which the phase characteristic is of the essence. A typical example are signals obtained from various sensors in industry. Therefore, it is necessary that a filter has linear phase characteristic to prevent loosing important information.
When a signal to be filtered is analysed in this way, it is easy to decide which type of digital filter is best to use. Accordingly, if the phase characteristic is of the essence, FIR filters should be used as they have linear phase characteristic. Such filters are of higher order and more complex, therefore. Otherwise, when it is only frequency response that matters, it is preferable to use IIR digital filters which have far lower order, i.e. are less complex, and thus much easier to realize.
The basic characteristics of Finite Impulse Response (FIR) filters are:
- linear phase characteristic;
- high filter order (more complex circuits); and
The basic characteristics of Infinite Impulse Response (IIR) are:
- non-linear phase characteristic;
- low filter order (less complex circuits); and
- resulting digital filter has the potential to become unstable.
