References of Nyquist Filter

7/21/2019 References of Nyquist Filter

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Abejo, Hanna E.Garcia, Jenica Anne M.Resurreccion, Christophe Rafael

Lth Band Filter (Lowpass)

The following list of references are gathered to provide knowledge andunderstanding regarding the Lth-band Filters or the Nyquist filters with a lowpassdesign. Based on the books we have collected, and several other articles on theinternet, we have found that the ideal filter design to use is the lowpass filter since anLth-band is ideally a lowpass filter. Presented with the list of references are the topics tobe studied and the concepts on how to design a Nyquist filter. On these references alsoare some example filter designs using lowpass Lth-Band filters, its characteristics, andsome concepts on how to learn about it.

We believe that the examples here would help us gain more information and help

us design a lowpass filter with the characteristics of an Lth-Band to complete our casestudy effectively.

References

1. Milic, Ljiljana. (2009). Multirate Filtering for Digital Signal Processing: MATLAB

Applications. Hershey, PA: Information Science Reference. Chapter 7, page 206-.

Lth Band Digital Filters

Chapter 7

Digital Lth-band FIR and IIR filters are the special classes of digital filters, which are of

particular interest both in single-rate and multirate signal processing. The common

characteristic of Lth-band lowpass filtersis that the 6 dB (or 3 dB) cutoff angular frequency is

located at /L, and the transition band is approximately symmetric around this frequency. In

time domain, the impulse response of an Lth-band digital filter has zero valued samples at the

multiples of L samples counted away from the central sample to the right and left directions.

Actually, an Lth-band filter has the zero crossings at the regular distance of L samples thus

satisfying the so-called zero intersymbol interference property. Sometimes the Lthband filters

are called the Nyquist filters.

The important benefit in applying Lth band FIR and IIR filters is the efficient

implementation, particularly in the case L = 2 when every second coefficient in the transfer

function is zero valued. Due to the zero intersymbol interference property, the Lth-band filters

are very important for digital communication transmission systems. Another application is the

construction of Hilbert transformers, which are used to generate the analytical signals. The Lth-

band filters are also used as prototypes in constructing critically sampled multichannel filter


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banks. They are very popular in the sampling rate alteration systems as well, where they are

used as decimation and interpolation filters in single-stage and multistage systems.

This chapter starts with the linear-phase Lth-band FIR filters. We introduce the main

definitions and present by means of examples the efficient polyphase implementation of the

Lth-band FIR filters. We discuss the properties of the separable (factorizable) linear-phase FIRfilter transfer function, and construct the minimum-phase and the maximum-phase FIR transfer

functions. In sequel, we present the design and efficient implementation of the halfband FIR

filters (L = 2). The class of IIR Lth-band and halfband filters is presented next. Particular

attention is addressed to the design and implementation of IIR halfband filters. Chapter

concludes with several MATLAB exercises for self study.

Lth-BAND LINEAR-PHASE FIR FILTERS: DEFINITIONS AND PROPERTIES

In this section, we consider the basic properties of the linear-phase Lth-band FIR filters. The

filter transfer function H(z) of such a filter can be expressed in the form

H(z)=

where, obviously, the filter length N is an odd number,

N = 2K +1.

Since the filter is of a linear phase, the impulse response coefficients are symmetric,

h[2K n]= h[n] for n = 0, 1, , 2K.

The frequency response of a linear-phase filter is expressible in the form

H()=H()where H() is the zero-phase frequency response given by

H()=

The filter H(z) is an Lth-band filter if the impulse response coefficients satisfy the following

conditions

h[K]=1 L, h[KrL]= 0 for r =1,2, , K L ,


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where [x] stands for the integer part of x. Figure 7.1(a) illustrates the above conditions for the

case K =10 and L = 4. Here, the value of the central coefficient h[10] is exactly 1/L = 1/4, and the

zero crossings occur at n = 10 4, and n = 10 8.

Equation (7.6) defines the time-domain conditions for the Lth-band filter. It was proved by

Mintzer(1982) that filters satisfying the time-domain conditions defined by (7.6) satisfy also thefollowing condition in the frequency domain

( )

MATLAB Functions

In MATLAB, function firnyquist from the Filter Design Toolbox designs a lowpass linear-

phase Lthband FIR filter with an equiripple magnitude characteristic. With the following code,

we compute the impulse response coefficients

h = firnyquist(Nord,L,ro);

Program firnyquist, for the given filter order Nord, factor L, and the roll-off factor ro,

returns in the vector h the impulse response coefficients of the linear-phase Lth-band filter. The

filter magnitude response exhibits equiripple characteristic in the pass and stopbands. Several

options are available with firnyquist, and some of them will be used later on in this chapter.

h = firnyquist(20,4,0.2); % Computing the Nyquist filter coefficients

Lth-band filters exhibit a very attractive property when they are used in interpolation.

Namely, when the interpolation-by-L is performed with an Lth-band filter, the original values of

the input samples appear at the output without any distortion at the regular time intervals of L

samples. The in-between L1 samples are determined by interpolation. This becomes evident

when observing the interpolation process in detail. It is well known that the interpolation- by-L

consists of two operations: up-sampling-by-L, and the lowpass filtering.

The illustrative example plotted in Figure 7.1 was designed by means of firnyquist with

the following parameters


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2. T. Saramaki, Handbook for Digital Signal Processing, Eds. S.K. Mitra, J.F. Kaiser, Wiley,

New York 1993.

4-10-2 Design of Lth Band (Nyquist) Filters


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3. Crochiere, Ronald and Rabiner, L. (1983). Multirate Digital Signal Processing. Murray Hill,

New Jesery: Prentice Hall.

Half-bands are a type of Lth-band filters (Nyquist filters) with an L value of two.


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4. Filter design toolbox for use with MATLAB. Users guide. Version 6. (2006). Natick:

MathWorks.

FIR Nyquist (L-th band) Filter Design

This example shows how to design lowpass FIR Nyquist filters. It also compares these filters

with raised cosine and square root raised cosine filters. These filters are widely used in pulse-

shaping for digital transmission systems. They also find application in interpolation/decimation

and filter banks.

Magnitude Response Comparison

The plot shows the magnitude response of an equiripple Nyquist filter and a raised cosine filter. Both filters have an

order of 60 and a rolloff-factor of 0.5. Because the equiripple filter has an optimal equiripple stopband, it has a larger

stopband attenuation for the same filter order and transition width. The raised-cosine filter is obtained by truncating

the analytical impulse response and it is not optimal in any sense.

NBand = 4;

N = 60; % Filter order

R = 0.5; % Rolloff factor

TW = R/(NBand/2); % Transition Bandwidth

f1 = fdesign.nyquist(NBand,'N,TW',N,TW);

f2 = fdesign.pulseshaping(NBand,'Raised Cosine','N,Beta',N,R);

heq = design(f1,'equiripple','Zerophase',true,'SystemObject',true);

hrc = design(f2,'window','SystemObject',true);

hfvt = fvtool(heq,hrc,'Color','white');

legend(hfvt,'Equiripple NYQUIST design','Raised Cosine design');


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In fact, in this example it is necessary to increase the order of the raised-cosine design to about 1400 in order toattain similar attenuation.

Impulse Response Comparison

Here we compare the impulse responses. Notice that the impulse response in both cases is zero every 4th sample

(except for the middle sample). Nyquist filters are also known as L-th band filters, because the cutoff frequency is Pi/L

and the impulse response is zero every L-th sample. In this case we have 4th band filters.

f1.FilterOrder = 38;

f2.FilterOrder = 38;

h1 = design(f1,'equiripple','Zerophase',true,'SystemObject',true);

h2 = design(f2,'window','SystemObject',true);hfvt = fvtool(h1,h2,'Color','white','Analysis','Impulse');

legend(hfvt,'Equiripple NYQUIST','Raised Cosine');

title('Impulse response, Order=38, Rolloff = 0.5');


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Nyquist Filters with a Sloped StopbandEquiripple designs allow for control of the slope of the stopband of the filter. For example, the following designs have

slopes of 0, 20, and 40 dB/(rad/sample)of attenuation:

set(f1,'FilterOrder',52,'Band',8,'TransitionWidth',.05);

h1 = design(f1,'equiripple','SystemObject',true);

h2 = design(f1,'equiripple','StopbandShape','linear','StopbandDecay',20,...

'SystemObject',true);

h3 = design(f1,'equiripple','StopbandShape','linear','StopbandDecay',40,...

'SystemObject',true);

hfvt = fvtool(h1,h2,h3,'Color','white');

legend(hfvt,'Slope=0','Slope=20','Slope=40')


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Minimum-Phase DesignWe can design a minimum-phase spectral factor of the overall Nyquist filter (a square-root in the frequency domain).

This spectral factor can be used in a similar manner to the square-root raised-cosine filter in matched filtering

applications. A square-root of the filter is placed on the transmiter's end and the other square root is placed at the

receiver's end.

set(f1,'FilterOrder',30,'Band',NBand,'TransitionWidth',TW);

h1 = design(f1,'equiripple','Minphase',true,'SystemObject',true);

f3 = fdesign.pulseshaping(NBand,'Square Root Raised Cosine','N,Beta',N,R);

h3 = design(f3,'window','SystemObject',true);

hfvt = fvtool(h1,h3,'Color','white');

legend(hfvt,'Minimum-phase equiripple design',...

'Square-root raised-cosine design');


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Decreasing the Rolloff FactorThe response of the raised-cosine filter improves as the rolloff factor decreases (shown here for rolloff = 0.2). This is

because of the narrow main lobe of the frequency response of a rectangular window that is used in the truncation of

the impulse response.

set(f1,'FilterOrder',N,'TransitionWidth',.1);

set(f2,'FilterOrder',N,'RolloffFactor',.2);

h1 = design(f1,'equiripple','Zerophase',true,'SystemObject',true);

h2 = design(f2,'window','SystemObject',true);

hfvt = fvtool(h1,h2,'Color','white');

legend(hfvt,'NYQUIST equiripple design','Raised Cosine design');


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Windowed-Impulse-Response Nyquist DesignNyquist filters can also be designed using the truncated-and-windowed impulse response method. This can be

another alternative to the raised-cosine design. For example we can use the Kaiser window method to design a filter

that meets the initial specs:

set(f1,'TransitionWidth',TW);

hwin = design(f1,'kaiserwin','SystemObject',true);

The Kaiser window design requires the same order (60) as the equiripple design to meet the specs. (Remember that

in contrast we required an extraordinary 1400th-order raised-cosine filter to meet the stopband spec.)

hfvt = fvtool(heq,hrc,hwin,'Color','white');

legend(hfvt,'Equiripple design',...

'Raised Cosine design','Kaiser window design');


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Nyquist Filters for InterpolationBesides digital data transmission, Nyquist filters are attractive for interpolation purposes. The reason is that every L

samples you have a zero sample (except for the middle sample) as mentioned before. There are two advantages to

this, both are obvious by looking at the polyphase representation.

fm = fdesign.interpolator(4,'nyquist');

Hm = design(fm,'kaiserwin','SystemObject',true);

hfvt = fvtool(Hm,'Color','white');

set(hfvt,'PolyphaseView','on');


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The polyphase subfilter #4 is an allpass filter, in fact it is a pure delay (select impulse response in FVTool, or look atthe filter coefficients in FVTool), so that: 1. All of its multipliers are zero except for one, leading to an efficient

implementation of that polyphase branch. 2. The input samples are passed through the interpolation filter without

modification, even though the filter is not ideal.


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5. Diniz, Paulo S. R.Adaptive Filtering: Algorithms and Practical Implementation. Springer Science& Business Media, 2012.

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References of Nyquist Filter