FILTER DESIGN IN DIGITAL SIGNAL PROCESSING Aravindha Karthik Chellappa; Srikanth Reddy Guntlakunta

ABSTRACT Filter Design is a vital part in Digital Signal Processing. The proper prototyping of a circuit (or, a structure) for filter design is very much essential for the system to work effectively and efficiently. The paper deals with an outline of the various filters, their design techniques, the efficiency of each one in real-time, and their applications.

An analog signal must be converted into digital form before Signal Processing techniques can be applied. For example, an analog electrical voltage signal can be digitized using an electronic circuit, popularly known as ADC or Analog-to-Digital Converter. This generates a digital output as a stream of binary numbers whose values represent the electrical voltage input to the device at each sampling instant.

Index Terms— Filtering, FIR Digital Filters, IIR Digital Filers 1. DSP – DEFINITION Digital Signal Processing is the processing of digitizing signals. A signal is a stream of information representing anything from basic prices in the market to data from a remote-sensing satellite. The term ‘digital’ comes from ‘digit’, meaning a number, so ‘digital’ literally means numerical. The French origin for digital is numerique. The term ‘digital’ and ‘discrete’ are often confused, for the exact opposite synonyms. The term ‘Digital’ means representation of signals in 0’s and 1’s, and ‘Discrete’ means the range of value of the digital signals after quantization. 2. FILTER DESIGN Filtering is the most common form of signal processing used in all the applications of DSP [1], to remove the frequencies in certain parts and to improve the magnitude, phase, or group delay in some other part(s) of the signal spectrum. Filters are designed based on: (1) The theory of approximation to derive the transfer function of the filter such that the magnitude, phase, or group delay approximates the given frequency response specifications and (2) Procedures to design the filters using the hardware components. 3. ANALOG AND DIGITAL SIGNAL PROCESSING Generally, the signal of interest is initially in the form of an analog-electrical voltage or current, produced, for example, by a microphone or some other type of transducer. In some situations, such as the output from the readout system of a Compact Disc player, the data is already in digital form.

Figure 3.1 Example of an analog signal processing system. Signals commonly need to be processed in a variety of ways. The output signal from a transducer may well be mixed with unwanted electrical ‘noise’. The electrodes attached to a patient’s chest when an ECG is taken, measure tiny electrical voltage changes due to the activity of the heart and the surrounding muscles. The signal is often strongly affected by ‘mains pickup’ due to electrical interference from the power supply.

Figure 3.2 Example of a digital signal processing system. Processing the signal using a filter circuit can remove, or at least, reduce the unwanted part of the signal. Increasingly nowadays, the filtering of signals to improve signal quality or to extract important information is done by Signal Processing techniques rather than by analog electronics. 4. DIGITAL FILTERS The basic elements in digital filters are the multipliers, adders, and delay elements, and they carry out multiplication, addition, and shifting operations on numbers according to an algorithm determined by the transfer function of the filters or their equivalent models.

They provide more flexibility and versatility compared to analog filters. The coefficients of the transfer function and the sample values of the input signal can be stored in the memory of the digital filter hardware or on the computer (PC, workstation, or the mainframe computer), and by changing the coefficients, we can change the transfer function of the filter, while changing the sample values of the input, we can find the response of the filter due to any number of input signals. This flexibility is not easily available in analog filters. The digital filters are easily programmed to do time-shared filtering under time division multiplexing scheme, whereas the analog signals cannot be interleaved between timeslots. Digital filters can be designed to serve as time-varying filters also by changing the sampling frequency and by changing the coefficients as a function of time, namely, by changing the algorithm accordingly. The digital filters have the advantage of high precision and reliability. Very high precision can be obtained by increasing the number of bits to represent the coefficients of the filter transfer function and the values of the input signal. Again we can increase the dynamic range of the signals and transfer function coefficients by choosing floating-point representation of binary numbers. The values of the inductors, capacitors, CMOS transistors used in the analog filters cannot achieve such high precision. Even if the analog elements can be obtained with high accuracy, they are subject to great drift in their value due to manufacturing tolerance, temperature, humidity—depending on the type of device technology used—over long periods of service, and hence their filter response degrades slowly and eventually fails to meet the specifications. In the case of digital filters, such effects are nonexistent because the word-length of the transfer coefficients as well as the product of addition and multiplication within the filter do not change with respect to time or any of the environmental conditions that plague the analog circuits. Consequently, the reliability of digital filters is much higher than that of analog filters, and this means that they are more economical in application. 5. FILTER TYPES AND DESIGN METHODS The different types of filters and methodologies for Chebyshev and Butterworth polynomial realisation are: [1] FIR [2] filters (Gibbs’ Phenomenon), and, [2] IIR [3] filters (Yule Walker Approximation)

5.1 Finite Impulse Response (IIR) Filters A Finite Impulse Response Digital Filter, usually consisting only of Zeros (no Poles), and generally implemented by a fixed point DSP processor to produce at low cost, Equiripple digital filters. The impulse response is "finite" because there is no feedback in the filter; if you put in an impulse (that is, a single "1" sample followed by many "0" samples), zeroes will eventually come out after the "1" sample has made its way in the delay line past all the coefficients. The transfer function of an FIR filter, in particular, is given by: and the difference equation describing this FIR filter is given by

The output due to the unit sample function δ(n)is the unit sample response or the unit impulse response denoted by h(n). So the samples of the unit impulse response h(n) = b(n). That is why the system is called the finite impulse response filter or the FIR filter. It has also been known by other names such as the transversal filter, non recursive filter, moving-average filter, and tapped delay filter. Since h(n) = b(n) in the case of an FIR filter, we can represent it by:

5.1.1 Types and Design of FIR Filters There are four types of FIR filters available – Type I, Type II, Type III and Type IV. Type I filters have a nonzero magnitude at ω = 0 and also a nonzero value at the normalized frequency ω/π = 1 (which corresponds to the Nyquist frequency), whereas type II filters have nonzero magnitude at ω = 0 but a zero value at the Nyquist frequency. So it is obvious that these filters are not suitable for designing bandpass and high pass filters, whereas both of them are suitable for low pass filters. The type III filters have zero magnitude at ω = 0 and also at ω/π = 1, so they are suitable for designing bandpass filters but not low pass and band stop filters. Type IV filters have zero magnitude at ω = 0 and a nonzero magnitude at ω/π = 1. They are not suitable for designing low pass and band stop filters but are candidates for band pass and high pass filters.

The magnitude response of the four types of FIR filters is given by:

5.2 Infinite Impulse Response (IIR) Filters Infinite Impulse Response Digital Filter consists of the equivalent of both poles and zeros, generally implemented with a floating point DSP [1] processor and capable of mathematically emulating most analog filters. The impulse response is "infinite" because there is feedback in the filter; if you put in an impulse (a single "1" sample followed by many "0" samples), an infinite number of non-zero values will come out (theoretically).

Figure 5.1.1.1 Magnitude Response of FIR filter (a) Type I; (b) Type II; (c) Type III; (d) Type IV 5.1.2 Gibbs Phenomenon These are some of the features of what is known as the “Gibbs phenomenon,” which was mathematically derived by Gibbs. We explain it qualitatively as follows. The finite sequence c(n); −M ≤ n ≤ M can be considered as the result of multiplying the infinite sequence c(n); −∞ ≤ n≤∞ by a finite window function:

Usually the specifications for a digital filter are given in terms of normalized frequencies. In many applications, the specifications for an analog filter are realized by a digital filter in the combination of an ADC [5] (at the transmitter) and a DAC [6] (at the receiver), and these specifications will be in the analog domain. 5.2.1 Design of Prototypes and Impulse Response If we are given the input–output sequence, it is easy to find the transfer function H(z) as the ratio of the z transform of the output to the z transform of the input. If, however, the frequency response of the system is specified, in the form of a plot, such as when the pass band and stop band frequencies along with the magnitude and phase over these bands, and the tolerances allowed for these specifications, are specified, finding the transfer function from such specifications is based on approximation theory. The transfer function of the IIR filter is given by

So we have the product hw(n) = c(n) · wR(n), which is of finite length as shown in Figure 5.1.2.1.

The Frequency response of a low pass filter, showing Gibbs overshoot is given below:

Figure 5.1.2.1 Frequency Response of Low Pass Filter, with Gibb’s Overshoot

Designing an IIR filter usually means that we find a transfer function H(z) in the form of above equation such that its magnitude response (or the phase response, the group delay, or both the magnitude and group delay) approximates the specified magnitude response in terms of a certain criterion. When we consider a few properties of transfer function when it is evaluated on a unit circle z=(ejω), the magnitude of the output signal is multiplied by the magnitude │H(ejω)│and the phase is increased by θ(ejw), and the response is given by:

The magnitude response of an IIR Filter is given below: specifications: maximum gain = 5 Db [6]; cutoff frequency = 1000 rad/s [7], with gain = 2 dB; stop band frequency = 5000 rad/s, with magnitude less than −25 dB.

has the form [num,den] = yulewalk(N,F,D) where F is a vector of discrete frequencies in the range between 0 and 1.0, where 1.0 represents the Nyquist frequency; the vector F must include 0 and 1.0. The vector D contains the desired magnitudes at the frequencies in the vector F; hence the two vectors have the same length. N is the order of the filter. 6. ADVANTAGES OF FIR FILTERS OVER IIR FILTERS

Fig 5.2.1.1 Magnitude response specifications of prototype IIR filters: (a) Butterworth filter; (b) Chebyshev (equiripple) filter. 5.2.2 Yule-Walker Approximation This function, called yulewalk, is used to find an IIR filter that approximates an arbitrary magnitude response. The method minimizes the error between the desired magnitude represented by a vector D and the magnitude of the IIR filter H(ejω) in the least-squares sense. In addition to the maximally flat approximation and the minimax (Chebyshev or equiripple) approximation we have discussed so far, there is the leastsquares approximation, which is used extensively in the design of filters as well as other systems. The error that is minimized in a more general case is known as the least-pth approximation. It is defined by:

The FIR filters have a few advantages over the IIR filters: (1) A FIR filter with a constant group delay is highly desirable in the transmission of digital signals. (2) The samples of its unit impulse response are the same as the coefficients of the transfer function. (3) The FIR filters are always stable and are free from limit cycles that arise as a result of finite word length representation of multiplier constants and signal values. (4) The effect of finite word length on the specified frequency response or the time-domain response or the output noise is smaller than that for IIR filters. 7. FILTERS IN DSP Filter Design methodologies of Digital signal processing is used in several areas, including the following: Telecommunications, Speech Processing, Consumer Electronics, Biomedical Systems, Image Processing, Military Electronics, Industrial Applications. 8. CONCLUSION

When p = 2, it is known as the least-squares approximation. In the error function shown above, D(ejω) is the desired frequency response and H(ejω) is the response of the filter designed, whereas W(ejω) is a weighting function chosen by the designer. It has been found that as p approaches ∞, the error is minimized in the minimax sense, and in practice, choosing p = 4,5,6 gives a good approximation to D(ejω) in the least-pth sense [14]. It is best to avoid sharp transitions in the specifications for the desired magnitude for the IIR filter when we use the MATLAB function yule walk. The function ----------------------------------------------------------------------[1] DSP – Digital Signal Processing [2] FIR – Finite Impulse Response [3] IIR – Infinite Impulse Response [4] ADC – Analog to Digital Converter [5] DAC – Digital to Analog Converter [6] dB – Decibel [7] rad/s – Radians per second [8] Hz - Hertz

In summary the existing methods of FIR and IIR filters was demonstrated and their usage in real-time applications was explained. The filter design methods have the advantages of making the receiver design simple, but still efficient in the presence of channel noise and in multipath environments. The applications of the various filters and their usage in the fields of audio and video signal transmission should be investigated in the future. 9. REFERENCES [1] S. K. MITRA AND K. HIRANO, Digital allpass networks, IEEE Trans. Circuits Syst,, 688-700 (1974). [2]SHENOI B.A, “Introduction to Digital Signal Processing and Filter Design”, Wiley-Interscience, 2005. [3] R. W. Hamming, Digital Filters, Prentice-Hall, 1977. [4] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, Prentice-Hall, 1989.