EE 3101留学生代做、代写Matlab编程、代做Computer Engineering、Matlab程序代写

日期：2018-11-30 10:22

Electrical and Computer Engineering Dept.

Dr. Monty A. Escabi

EE 3101, Fall 2018

MATLAB Application Project

Due: Dec. 7, 2018

3-Band Digital Audio Equalizer

The goal of this project is to build and test a digital audio equalization network in which you will be able to

boost or attenuate the low, mid and high frequency range of music sounds. Audible sounds cover a frequency

range from 20 Hz – 20 kHz (for human hearing). In many listening scenarios sounds are distorted by the audio

system (e.g., the speaker transfer function) or the acoustic environment (room transfer function). One can

correct for the magnitude of such distortions with an audio equalizer. High end audio equalizers typically have

~30 selectable bands that allow you to adjust the sound gain in 1/3 octave frequency steps for frequencies

between 20Hz – 20 kHz. For the purpose of this project, we will consider a simpler equalization network that

allows you to adjust the bass, midrange and treble bands.

In this project, you will design and build a three-band equalizer where you will be able to separately adjust the

bass, midrange and treble using MATLAB. The schematic for this design is illustrated below:

The input sound waveform, x(t), is decomposed by three filters and these outputs are routed a gain term for

each of the three bands (Gb=bass, Gm=mid, and Gt

=treble). The signals are then added together to generate the

corrected output.

The lowpass (LP) filter encompasses frequencies below 100 Hz (bass range). The midrange is selected with a

bandpass (BP) filter covering frequencies in the range 100Hz-5 kHz. The trebles is selected with a highpass

filter (HP) that encompasses the remainder of the audible range (5-22.05 kHz).

You are allowed to work in groups of two or three. However, each student is required to hand in their own

independent report and is required to implement his/her own code. Reports using copied code or text will

be returned without a grade. (Yes, we will check for this! Code needs to be uploaded onto HUSKYCT). Group

members who worked on the project should be identified in the writeup.

Filter Design

HP

BP

LP

x(t)

GB

GM

GT

Treble

Midrange

Bass

+ y(t)

Each of the above filters is implemented using a Kaiser window FIR (Finite Impulse Response) filter

(see chapter 4.9 & 12.8-1 for windowing and FIR filter design). In general, windowed FIR filters are

implemented as the product of an ideal filter impulse response ( hideal [k], for a lowpass, bandpass or highpass)

and a window function (w[k] ) that is used to smoothly truncate the ideal filter impulse response to a total of

2N+1 samples

h[k] = hideal [k]?w[k]

Here N is a filter parameter referred to as the filter order where the total number of coefficients in the discrete

time impulse response of the filter is 2N+1. Conceptually, note that any “ideal” filter requires an infinite amount

of time and therefore an infinite amount of coefficients to implement. As we have discussed in class, this is not

practically feasibleand the ideal filters cannot be implemented in practice. To overcome this, the window

function is used to truncate the filter in a smooth fashion in order to 1) require only finite number of time

samples (2N+1) and 2) minimize filter distortions (or errors) in the passband and stopband.

There are a variety of window functions that can be used to generate an FIR filter as above. In this

project we will be using a Kaiser window which is widely used for singal processing. The Kaiser window

contains a single parameter ( β ) that is used to control the filter “smoothness” in the time domain. In the

frequency domain this parameter serves to adjust the passband and stopband errors.

In the above equation, each of the ideal filter impulse responses are given by

Lowpass

ideal

2p sinc 2

where Fc is the cutoff frequency of the ideal lowpass filter and Fs =44100 Hz is the sampling rate of the data.

Bandpass

"#$%& = 2 *+

,-

*+

,-

cos 2 89 +/2 />

where Fc1 and Fc2 are the lower and upper cutoff frequencies of the ideal bandpass filter and BW = Fc2 Fc1 is

the filter bandwidth.

Highpass

hideal [k] = δ [k] 2Fc

Fs

sinc

2πFc

Fs

k "

#

$ %

&

'

where Fc is the ideal highpass filter cutoff frequency and δ [k] is the discrete time Diract delta function (i.e., 1

for k=0 and zero otherwise). Note that this highpass filter impulse response is simply the difference between an

allpass filter (δ [k]) and an ideal lowpass filter (the sinc function term).

In all of the above filters k spans the integers (from -∞ to ∞ ), however, in the actual filter

implementation described below the filters will be truncated from –N to N (for a total of 2N+1 coefficients).

Also, note that in MATLAB the sinc function is defined as sin(π x) /(π x) whereas above and in the book it is

defined as sin(x)/x. You will need to remove π from the sinc function during implementation in MATLAB to

obtain the correct results.

Filter Implementation

To generate the Kaiser filter you will

1) Need to determine the Kaiser order (N) and smoothness parameter ( β ). As described in the class for the

Butterworth filter, the filter parameters are obtain with a mapping equation that converts the filter

specifications to filter parameters. For a Kaiser filter, there are two filter specifications of interest.

a) ATT = is the filter attenuation. This parameter corresponds to the amount of error in the

passband and stopband in units of dB. For Kaiser filters, the stopband and passband errors

are symmetric so we don’t need separate errors to describe each and we can use a single

number. For example, if the error in the passband is δ =0.01 (relative to a passband gain of

1), then ATT = ?20log10 (δ) = ?20 log10 (0.01) = 40 dB (in the graph below ATT=As). For the

case of a bandpass filter, two errors may be provided, one for the stopband and one for the

passband. Choose the more stringent of the two for the Kaiser filter design.

b) TW= is the transition width of the filter in units of Hz. Conceptually, TW is the frequency

range over which the filter transitions from the passband to the stopband (or vise versa for a

highpass filter). If fp is the passband frequency in Hz and fs is the stopband frequency, then

TW= fs - fp . The filter specifications are shown graphically below for a lowpass filter.

Note that the cutoff frequency is directly in the center between the stopband and passband

frequencies ( fc = fs + f ( p ) 2 ) (normalized to units of radians in the above graph). For the

case of a bandpass filter, two TW may be provided, one for the stopband and one for the

passband. Choose the more stringent of the two for the Kaiser filter design.

Given the above filter specification (ATT and TW or alternately the stopband and passband frequencies

and errors), you will obtain the filter parameters N and β using the following transformation

ROARK AND ESCAB′I: -SPLINE DESIGN OF MAXIMALLY FLAT AND PROLATE SPHEROIDAL-TYPE FIR FILTERS 703

for integer values of , where Equation

(5) for describes piecewise polynomials on the interval

and zero elsewhere, as indicated in Fig. 2.

The lowpass filter prototype is obtained by applying

(5) to the convolution integral for It is

straightforward to show that the result may be expressed as

for (6)

which describes a piecewise function that is nonzero on the

interval Equation (6) is valid for

and integer values of With

in (6) being a lowpass filter function, is recognized to

be the corresponding highpass filter function.

The discrete-time impulse response of the filter is obtained

by applying (3) to the integral formula (1) and using the

convolution theorem, resulting in

(7)

for , and It is noted that the magnitude

response in Fig. 1 and the impulse response in (7) approach

those of the rectangular filter as or as Impulse

response functions for other filter types, such as bandpass and

notch [16], may be developed in similar fashion.

Unlike most digital filters, these prototypes continue to

be unity throughout the passband and zero throughout the

stopband so that they are maximally flat for all derivatives

of with respect to for and

Further, is maximally flat at

in the sense that the first derivatives of are

zero there (as approached from ). Thus, the filter prototypes

have maximally flat prescriptions in both the passband and

stopband, in similar manner that prototypes are specified to

be maximally flat at and using the functions of

Hermann [1] in the FIR case or the polynomials of Butterworth

[17] for filters requiring rational function descriptions. The

original filter prototype, prior to convolving with , may

be any function for which it is desired to smooth transition

edges and not necessarily one with rectangular-like features.

The flatness features of are demonstrated in Appendix

A. A historical perspective of the evolvement of these filter

prototypes is provided in Appendix B.

III. PRINCIPALLY FLAT FILTERS

This manuscript follows the text of Mitra and Kaiser [7]

regarding definitions of cutoff frequency transition width

and stopband attenuation of the approximated frequency

response that results from truncation of the

convolution series, as depicted in Fig. 3.

The window method of FIR filter design maps target parameters

of the magnitude response to parameters of the impulse

response coefficients. For single-parameter window functions,

such as the Hamming, Hann, and Blackman windows, a

specification for transition width is mapped to a value

of (one half filter order). With these windows, there

Fig. 3. Definitions of stopband attenuation , transition width , and

cutoff frequency for the (real-valued) filter function associated with

the truncated convolution series (2).

are no additional parameters, and therefore no control is

provided over stopband attenuation or over the shape of

the magnitude response.

In 1974, Kaiser [3] introduced a window function that

approximated the prolate spheroidal wave function [18] while

being computationally easier to determine. Using the Kaiser

window, stopband attenuation of the magnitude response may

be controlled by mapping to a parameter of the window,

which is denoted by Kaiser as The Kaiser window filter

is a popular choice because it is relatively easy to design

and because many filtering applications require the highly

selective characteristics offered by a prolate spheroidal-type

window. However, the Kaiser window offers little control

otherwise over the basic shape of the filter magnitude response.

If control over shape of the magnitude response is desired (or,

equivalently, control over shape of the window), then another

parameter must be introduced into the window function.

Proposals of windows containing additional parameters have

met with mixed success [9], [10], [19]. Usually, the additional

parameter has been designed into the window function for

purposes of gaining a tighter transition width or increasing

stopband attenuation rather than to achieve other desirable

features, such as flatness in a selected region of the passband.

The filter introduced in this paper offers two parameters—

and —in addition to Therefore, specifications of and

can be expected to map to two of these parameters, whereas

the third may be used to control shape of the magnitude

response. Because the parameter in (7) represents the number

of convolutions or smoothing operations, it can be expected to

dominate control of amplitude error in the magnitude response.

Similarly, the parameter can be expected to dominate control

of transition width. Since the filter prototype has been

developed to be principally flat in the passband and stopband,

the filter magnitude response can be expected to offer

features of flatness as well.

A. The Need for Flat Passband Filters

There are many filtering applications in which passband

and stopband flatness are given equal or greater priority than

β =

0 ATT<21

0.5842(ATT ? 21)

0.4

+ 0.07886(ATT ? 21) 21≤ ATT ≤ 50

0.1102(ATT ?8.7) ATT > 50

#

$

%

%

&

%

%

N = ceil

Fs (ATT 7.95)

28.72 TW

#

$

% &

'

(

2) Once you obtain the filter parameters, the Kaiser window is obtain as a vector containing 2N+1 coefficients

using the following command in MATLAB:

w=kaiser(2*N+1,Beta);

You will also need to generate the desired ideal filter impulse as described above but truncated for a total of

2N+1 coefficients (i.e., k=-N … N). In MATLAB, you can assign the vectors h_low, h_band, h_high for

each of these filters.

If you generate the ideal filter impulse response as described above, the final filter is then generated as

h=h_ideal.*w;

where h_ideal is a vector containing the filter coefficients from the above equations.

Equalizer Specifications

The filters are designed to satisfy the following specifications.

Lowpass Filter:

Gp = 0.9 Gs = 0.01

?@ = 150 Hz > = 250 Hz

Bandpass Filter:

Gp1 = Gp2 = 0.9 Gs1 = Gs2 = 0.01

@9 = 150 Hz >9 = 250 Hz

@D = 4950 Hz >9D = 5050 Hz

Highpass Filter:

= 0.9 Gp Gs = 0.01

> = 4950 Hz @ = 5050 Hz

All sound waveform are sampled at a sampling rate of 44.1 kHz. Once you generate the Kaiser impulse

response as described above you can simply convolve the impulse response with the sound waveform to

generate the output:

y=conv(h,x);

where h is the impulse response vector, x is the input sound vector, and y is the output sound vector.

Simulation

You are to simulate the crossover filtering procedure for two separate signals.

1) White noise – similar to the hissing noise you hear when your FM radio is not tuned to a station

2) Music sound sample provided

I will provide several short music sound sample as a MATLAB data file (to load type: ‘load filename.mat’). To

generate a sample of white noise simply use the command:

x=randn(1,44100*10);

This generates a 10 second white noise signal.

Note that once your code is written for white noise you simply have to change the array x to resimulate the

equalization network for music. No additional work is necessary.

For both signals (white noise & music sample), you will simulate three versions of the equalization filterbank.

These include:

1) 20 dB gain in the bass section with the other two bands set to 0 dB gain.

2) 20 dB gain in the midrange section with the other two bands set to 0 dB gain.

3) 20 dB gain in the treble section with the other two bands set to 0 dB gain.

Plotting Results

You are required to plot the results both in the time and frequency domain for both the input and the output. To

plot the time-waveforms simply use the command:

plot(taxis,x)

where taxis is a time axis array containing the sample time points and x is an array containing the waveform. To

plot the magnitude spectrum of the input or output signals simply use the power spectral density command

(pwelch; type ‘help pwelch’ for details). The syntax will typically be something as follows:

pwelch(X,512,[],[],Fs)

where Fs=44100 is the sampling rate.

Note that there are a total of 12 graphs to plot (2 sounds x 3 filters/sound * (1 time domain graph/filter + 1 freq.

domain graph/filter) = 12 graphs)

Generating WAV Sound File and playing the sound:

You are expected to listen to each of the sound files at each filter output stage of the crossover so that you gain

some insight into the transformations that are being performed at each stage.

You can also listen to the sound using the SOUNDSC command. For the sound vector X, you can play the

sound using the syntax: soundsc(X,Fs).

You can convert each data array into a WAV sound file. You can do this with the audiowrite command in

MATLAB. The syntax for the command is as follows:

audiowrite(wavfile,Ys,Fs,’BitsPerSample’,nbits)

where

Ys=Y/max(abs(Y)) is the rescaled signal vector

Fs= is the sampling frequency in Hz

nbits=is the number of bits (i.e. The resolution) of each sample

wavfile=is the output file name

For additional information in MATLAB you can type: help audiowrite

For this project, you will use the parameters

Fs=44100

nbits=16

Let’s say you want to write the data of an array X to a WAV file named: test.wav. You can use the syntax:

audiowrite(‘test.wav’,X/max(abs(X)),Fs,’BitsPerSample’,nbits);

Writeup and submission

A short writeup describing the methods and results is required. At minimal it should include.

1) A brief introduction describing the project objective (~ 1 page).

2) A brief description of the methods (~1-2 pages).

3) A discussion of the results. Focus on describing:

a) How the time waveform changes between input and output or both music and white noise.

b) How each equalization network changes the magnitude spectrum of music and white noise.

c) Compare and contrast the input and output waveforms.

d) Listen to the input and output sounds using the soundsc routine and compare and contrast the

audible differences between the input and output for all of the different conditions.

4) Program code – uploaded to Husky CT as a single file “Project3101_LastName.m”

5) Figures – time waveform and power spectrum of input and output for each condition.

6) Wav Sounds – You’ll need to upload the wav sounds for the three networks (bass, mid and treble

versions) and two sound versions (Music or white noise, WN) with the following file names

BassMusic_LastName.wav

TrebleMusic_LastName.wav

MidMusic_LastName.wav

BassWN_LastName.wav

TrebleWN_LastName.wav

MidWN_LastName.wav

Honors Student Requirements

In addition to implementing the equalization network using the Kaiser filters, also implement the above network

(identical passband and stopband parameters) using a Butterworth filter. Details for the implementation of the

Butterworth filters in MATLAB can be found in section 7.5 of the book (e.g., see example C7.5). As for the

Kaiser filter, plot the same set of results (time and frequency domain), but only do so for white noise.

NOTE: When implementing the analog Butterworth filters using the transfer function syntax as described in the

book, you might run into some numerical stability issues. To overcome these issues, you will need to determine

the normalized passband (Wp) and stopband (Ws) frequencies (normalized by the Nyquist frequency) and

design a digital Butterworth filter using the zeros, poles, and gain syntax. For example, to design a digital

lowpass filter for given Gp, Gs (in dB), Wp, Ws use the following matlab commands

[n,Wc] = buttord(Wp, Ws, Gp, Gs);

[z,p,k] = butter(n,Wc);

sos = zp2sos(z,p,k);

Y = sosfilt(sos,X);

Where X is your input sound vector.

a) Contrast and compare the filters (Kaiser vs. Butterworth). Do they sound similar or are there audible

differences when you implement the crossover for the music. Visually, do the spectrums look similar or

different when you use white noise. Describe any discrepancies or similarities.