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MEE 0828

SOFTWARE IMPLEMENTATION

OF DIGITAL FILTERS

Satish kumar Are

Manoranjan Reddy Thangalla

Saikrishna Gajjala

This thesis is presented as part of Degree of

Master of Science in Electrical Engineering

Blekinge Institute of Technology

August 2008

Blekinge Institute of Technology

School of Engineering

Department of Signal Processing

Supervisors: Nedelko Grbic

Mikael Swartling

!"342!#4 This thesis proposes to create a MATLAB GUI (Graphical User Interface) to replace an existing laboration exercise in signal processing at Blekinge Institute of Technology. MATLAB is a matrix-based technical computing language widely used throughout the scientific, engineering and mathematical communities. A GUI provides a graphical interface between the program and the user, facilitating ease and frequency of use. Development of a MATLAB GUI for this laboration exercise will benefit the students and increase the awareness towards designing of digital filters. The developed software provides an interface between audio recording and playback hardware and the user when exploring filter design parameters. This software is designed for analyzing digital filter characteristics such as amplitude, phase and pole/zero locations which are useful in designing an appropriate filter. This can be achieved by entering arbitrary filter parameters. !#+./7,%$'-%.43 We would like to express our sincerest thanks and gratitude to Mikael Swartling for being our mentor on this journey. His guidance, patience and support throughout this project have been a blessing. We would also like to thank our teacher of signal processing Nedelko Grbic for providing such a good project and interesting lectures on signal processing. Finally we would like to thank Mikael Åsman for support and suggestions towards the project. For analysis of digital filters, DOS based software was developed years back at Blekinge Institute of Technology for students of signal processing. However, this software have drawbacks such as portability, inefficient use of computer resources, less accessibility and bulk hardware system due to floating point DSP which requires extra hardware for floating point operations. The developed software has some constraints as it is not user friendly. With extensive advancements in scientific software, a user friendly graphical user interface was developed using MATLAB graphical user interface for the analysis of digital filters. This GUI will overcome the complexities mentioned earlier. The developed software will help the user to analyze the filters in an efficient manner due to the availability of input signals, windows, various types of filters such as lowpass, highpass, bandpass, and bandstop filters, and pole/zero plot with filter coefficients.

Chapter 1: Digital Filters

1.1 Background

The term 'filter' is frequently used in signal processing. A filter is a frequency selective device that removes unwanted information from the original message signal. Unwanted signals can be noise or other undesired information. Digital filters are more versatile when compared to the analog filters in their characteristics such as programming flexibility, ability to handle both low as well as high frequency signals accurately. Also the hardware requirement is relatively simple and compact. In real world signals are analog in nature. A simple signal flow block diagram that explains how the signal is processed to acquire desired output signal is shown in figure 1.1. unfiltered sampled digitally filtered analog digitised filtered analog signal signal signal signal

Figure 1.1: Signal flow block diagram.

Analog to digital conversion is an engineering process that enables digital processor to interact with real world signals. The input to the processor should be properly sampled and quantized. Sampling and quantization restrict the amount of information a digital signal contain. In the figure 1.1 an interface is provided between analog signal and the digital signal processor called analog to digital converter (ADC). The output from ADC is input to the processor. In applications output from the processor is to be given to user in analog form such as speech communications, for this an interface is provided from digital domain to the analog domain. This interface is called digital to analog converter (DAC). Thus the signal is provided in analog for to the user as shown in figure 1.1. The processor in figure 1.1 can be anywhere from a large programmable digital computer to a small microprocessor which contains digital filters. ADC DAC

PROCESSOR

The digital filters are two types based on their impulse response; finite impulse response (FIR) and infinite impulse response (IIR) filters. FIR filters have same time delay for all frequencies (linear phase), relatively insensitive to quantization and are always stable. FIR filters can be designed in different ways, for example window method, frequency sampling method, weighted least squares method, minimax method and equiripple method. Out of these methods, the window technique is most conventional method for designing FIR filters.

1.2 FIR filters

A finite impulse response filter of length ࢖ with input ࢻ቗ࢱቘ and output ࢼ቗ࢱቘ is described by

the difference equation where

ࢥࢮ is the set of filter coefficients. The transfer function of this filter in ࢽ domain can be

represented as A window in filter design provides trade off between resolution that is the width of the peak and spectral leakage that is the amplitude of the tails of desired impulse response. The desired frequency response specification for linear phase filter is the Fourier transform of the desired impulse response, and this can be represented as and the inverse as where

࢑ࢧ቗ࣰቘ is the desired frequency response and ࢫࢧ቗ࢱቘ is the corresponding impulse

response. As

ࢫࢧ቗ࢱቘ is infinite duration, the sample response must be truncated. Truncation is

performed by multiplying desired sample response with a window function in time domain which gives sample response of filter represented as where

ࢺ቗ࢱቘ is a window function. Various types of windows were used when designing the

FIR filters.

The rectangular window has excellent resolution characteristics for signals of comparable strength. The rectangular window is defined as The frequency response of the window function is the Fourier transform which, is defined as The amplitude response of the rectangular window function is and the phase response is The actual impulse response can be expressed in frequency domain as convolution which leads to smoothing of

࢑ࢧ቗ࣰቘ. As ࢖ increases, ࢠ቗ࣰቘ becomes narrower, thereby reducing

the smoothing effect. In figure 1.2 it is observed that as ࢖ increases, the main lobe becomes narrower. However, the amplitude of the side lobes is unaffected. The frequency response of a lowpass FIR filter designed using rectangular window is shown in figure 1.3 with cutoff frequency for different window lengths, where cutoff frequency is the characteristic frequency which determines the type of the filter. Figure 1.2: Frequency response for Rectangular window. Figure 1.3: Lowpass FIR filter designed with Rectangular Window. A Bartlett window is a triangular shaped window function. The Bartlett window has higher side lobe attenuation than the rectangular window. The Bartlett window is defined as The frequency response for Bartlett window is shown in figure 1.4 and figure 1.5 shows the frequency response of a lowpass FIR filter designed using Bartlett window.

00.050.10.150.20.250.30.350.40.450.5-150

-100 -50 0

Normalized frequency

Magnitude(dB)

M=9 M=15 M=21

00.050.10.150.20.250.30.350.40.450.5

-140 -120 -100 -80 -60 -40 -20 0 20

Normalized frequency

Magnitude(dB)

M=9 M=15 M=21 Figure 1.4: Frequency response for Bartlett window. Figure 1.5: Lowpass FIR filter designed with Bartlett Window. The Hanning window is a raised cosine window and can be used to reduce the side lobes while preserving a good frequency resolution compared to the rectangular window. It is commonly used as general purpose window for the analysis of continuous signals. The

Hanning window is defined as

00.050.10.150.20.250.30.350.40.450.5-150

-100 -50 0

Normalized frequency

Magnitude(dB)

M=9 M=15 M=21

00.050.10.150.20.250.30.350.40.450.5

-140 -120 -100 -80 -60 -40 -20 0 20

Normalized frequency

Magnitude(dB)

M=9 M=15 M=21 The frequency response for Hanning window is shown in figure 1.6 and figure 1.7 shows the frequency response of a lowpass FIR filter designed using Hanning window. Figure 1.6: Frequency response for Hanning window. Figure 1.7: Lowpass FIR filter designed with Hanning Window. The Hamming window is, like the Hanning window, also a raised cosine window. The Hamming window exhibits similar characteristics to the Hanning window but further suppress the first side lobe. The Hamming window is defined as

00.050.10.150.20.250.30.350.40.450.5-150

-100 -50 0

Normalized frequency

Magnitude(dB)

M=9 M=15 M=21

00.050.10.150.20.250.30.350.40.450.5

-140 -120 -100 -80 -60 -40 -20 0 20

Normalized frequency

Magnitude(dB)

M=9 M=15 M=21 The frequency response for Hamming window is shown in figure 1.8 and figure 1.9 shows the frequency response of a lowpass FIR filter designed using Hamming window. Figure 1.8: Frequency response for Hamming window. Figure 1.9: Lowpass FIR filter designed with Hamming Window. The Blackman window is similar to the Hanning and the Hamming windows. An advantage with the Blackman window over other windows is that it has better stopband attenuation and with less passband ripple. The Blackman window is defined as

00.050.10.150.20.250.30.350.40.450.5-150

-100 -50 0

Normalized frequency

Magnitude(dB)

M=9 M=15 M=21

00.050.10.150.20.250.30.350.40.450.5

-140 -120 -100 -80 -60 -40 -20 0 20

Normalized frequency

Magnitude(dB)

M=9 M=15 M=21 The frequency response for Blackman window is shown in figure 1.10 and figure 1.11 shows the frequency response of a lowpass FIR filter designed using Blackman window. Figure 1.10: Frequency response for Blackman window. Figure 1.11: Lowpass FIR filter designed with Blackman Window. A design consideration when designing digital FIR filter is selecting a window. This can be done with the help of frequency specifications of the required filter. In general, the frequency

00.050.10.150.20.250.30.350.40.450.5-150

-100 -50 0

Normalized frequency

Magnitude(dB)

M=9 M=15 M=21

00.050.10.150.20.250.30.350.40.450.5

-140 -120 -100 -80 -60 -40 -20 0 20

Normalized frequency

Magnitude(dB)

M=9 M=15 M=21 specification consists of pass and stopband cutoff frequencies and attenuations. The length of the filter can be determined by the main lobe width. Table 1 shows the side lobe attenuation and main lobe width for different windows. Table 2 shows the desired impulse response functions for various filters. Specifically, stopband attenuation provides for a user to select an appropriate window. Window Side lobe attenuation Approximate Main lobe width

Rectangular -20dB Γࠅ݌⁄

Bartlett -27dB Ηࠅ݌⁄

Hanning -40dB Ηࠅ݌⁄

Hamming -50dB Ηࠅ݌⁄

Blackman -70dB ΐΑࠅ݌⁄

Table 1: Comparison of main lobe width and side lobe attenuation for different window types.

Filter Type Desired impulse response ܭ

Lowpass ࠎ௳

Highpass

Bandpass

Bandstop

Table 2: Desired impulse responses for filter types. From the window frequency response plots shown in the Figures 1.2, 1.4, 1.6, 1.8 and 1.10, one can observe that as M increases, the main lobe becomes narrower, side lobe amplitudes remain unaffected but width of the sidelobes decreases. The rectangular window provides less width in mainlobe and higher sidelobes in contrast with other windows. Using the window function the ringing effects at the band edges vanishes which results in lower sidelobes, thereby increase in the width of the transition band of the lowpass FIR filter as shown in Figures 1.3, 1.5, 1.7, 1.9 and 1.11. Impulse response functions of IIR filters are non-zero over an infinite length of time. IIR filters can be described using a difference equation as where expressed as These filters can be designed by the bilinear transformation method. These filters are designed using their analog counterparts rather than discrete time analysis. The bilinear transformation method is commonly used in designing digital IIR filters to obtain filter coefficients. As mentioned in section 1.6, digital filters are designed with their analog counterparts, so it must be transformed into discrete time domain. This transformation can be done with the bilinear transform. The amplitude response of a Butterworth filter is given as where

ࢗ is order of the filter, Ωࢳ is the passband frequency, Ω is the analog frequency of the

filter specifications, andquotesdbs_dbs14.pdfusesText_20

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