Pamantasan ng Lungsod ng MaynilaUniversity of the City of ManilaGen. Luna, Intramuros, Manila

College of Engineering and TechnologyComputer Engineering Department

BS CpE IV / 1st SemesterS.Y. 2015 – 2016

Digital Signal Processing CpE 415-1

Review of Related LiteratureLth Band Filter

CABAL, Roemil M.GERCAYO, Krishia P.

MERCADO, John LesterROMERO, Mhica Angel M.YAMSON, Eirry Rose Anne

Engr. Reynaldo Ted PeñasSeptember 03, 2015

Review of Related Literature

Fundamentals of Filter

a. Formula

Digital Lth-band FIR filter are the special classes of digital filters, which are of particular

interest both in single-rate and multi-rate signal processing. The filter transfer function H(z) of

an Lth band filter is expressed in the form:

H(z) = ∑n=0

2 K

h[n] z−n [Eq.1]

And its filter length is described by:

N = 2K+1 [Eq.2]

Where:

H(z) is the transfer function of the filter

N is the length of the filter

K is the desired input to vary the length of the filter

The common characteristic of Lth-band low-pass filters is that the cutoff angular

frequency is located at πL

. An Lth-band filter has the zero crossings at the regular distance of L

samples thus satisfying the so called zero inter-symbol interference property.

ωc = πL [Eq.3]

Where:

ωc is the cutoff angular frequency and L is the samples of zero crossings

The passband (ωp) and stopband (ωs) edge frequency are symmetric around cutoff

angular frequency (ωc). Passband and stopband frequencies are express by using a roll-off factor

(ρ) that ranges from 0<ρ<1.

ωp¿(1− ρ) πL [Eq.4]

ωs = (1+ρ)πL [Eq.5]

Where:

ωp is the passband edge frequency

ωs is the stopband edge frequency

ρ is the roll-off factor with a range of 0<ρ<1

In between the passband and stopband edge frequency is the transition band or transition width (tw) that is given by the equation:

tw = ρπL [Eq.6]

Another specifications for an ideal low pass filter that must be considered are: the peak

passband ripple (δp) and the peak stopband ripple (δs). One must understand that these two are

directly related to one another and is given by the equation:

δp=(L−1)δs [Eq.7]

When impulse response and frequency response is available, we can get the magnitude in

decibels simply with the formula:

20log(H(f)) [Eq.8]

The graphical representation of the above specifications may be seen in the figure below.

Figure 1: Ideal Low Pass Filter with specifications of passband (ωp), stopband (ωs), transitionband (tw) and cutoff angular frequency (

b.

Type

The term digital filter, or simply filter, is often used to refer to a discrete-time system. A

digital filter maybe defined as computational process or algorithm by which a sampled signal or

sequence of numbers (acting as the input) is transformed into a second sequence of numbers

termed the output signal. The computational process may be that of lowpass filtering

(smoothing), bandpass filtering, interpolation, the generation of derivatives, etc.

FIR filters are one of two primary types of digital filters used in Digital Signal Processing

(DSP) applications, the other type being IIR. "FIR" means "Finite Impulse Response". If you put

in an impulse, that is, a single "1" sample followed by many "0" samples, zeroes will come out

after the "1" sample has made its way through the delay line of the filter. In the common case,

the impulse response is finite because there is no feedback in the FIR. A lack of feedback

guarantees that the impulse response will be finite. Therefore, the term "finite impulse response"

is nearly synonymous with "no feedback". However, if feedback is employed yet the impulse

response is finite, the filter still is a FIR. An example is the moving average filter, in which the

nth prior sample is subtracted (fed back) each time a new sample comes in. This filter has a finite

impulse response even though it uses feedback: after N samples of an impulse, the output will

always be zero.

There are many special types of filter in FIR filter. Some of them are boxcar, Hilbert

transform, differentiator etc. For this project, again we used an Lth-Band Filter. Also called

"Nyquist" filters, these filters are a special class of filters used primarily in multi-rate

applications. Their key selling point is that one of every L coefficients is zero--a fact which can

be exploited to reduce the number of multiply-accumulate operations required to implement the

filter. (The famous "half-band" filter is actually an Lth-band filter, with L=2.)

Finally, an FIR Lth-Band Filter will be implemented in a low pass type of filter. A low-

pass filter is a filter that passes signals with a frequency lower than a certain cutoff

frequency and attenuates signals with frequencies higher than the cutoff frequency. The amount

of attenuation for each frequency depends on the filter design in this case, an FIR Lth Band

Filter. For the said type of filter, the only signal that can pass for a certain input is those signal

that is lower than πL

.

A typical FIR Lth Band Low Pass Filter is graphically represented below

Figure 2. An illustration of an Lth Band Low Pass Filter with L=4, K=10

(a) Impulse response (b) Magnitude response

Applications

The widespread use of digital representation of signals for transmission and storage has

created challenges in the area of digital signal processing. Digital Signal Processing has become

essential to the design and implementation of high performance audio, video, multi-media, and

communication systems signal processing. An essential component of cost effective DSP

algorithms is multi-rate signal processing.

Multirate signal processing techniques are widely used in many areas of modern

engineering such as communications, image processing, digital audio, and multimedia. The main

advantage of a multirate system is the substantial decrease of computational complexity, and

consequently, the cost reduction. The computational efficiency of multirate algorithms is based

on the ability to use simultaneously different sampling rates in the different parts of the system.

The multirate filtering makes the general concept of multirate signal processing applicable in

practice.

A multirate filter can be defined as a digital filter in which the sampling rate of the input

signal is changed in one or more intermediate points. Multirate techniques are used in filters for

sampling rate conversion where the input and output rates are different, and also in constructing

filters with equal input and output rates. An example of a filter that are primarily used for

multirate filtering is the Lth Band filters.

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

interest both in single-rate and multirate signal processing. Due to the zero intersymbol

interference property, the Lth-band filters are very important for digital communication

transmission systems. The Lth-band filters are also used as prototypes in constructing critically

sampled multichannel filter 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.

Since Lth Band Filters are widely used in the modern era, the researchers decided to state

its different effects and application in engineering areas such as digital audio and image

processing.

A. Audio

Based on the study done by Christopher Hicks, the importance of dithering in digital

audio systems has been recognized, at least on an academic level, for some time. Despite this, the

subject remains poorly understood in the audio profession at large, and the unpopularity of many

early digital recordings can be attributed to signal distortion at low levels, as a result of

unsuitable or non-existent dither. The technique of noise-shaping has been applied to digital

audio in a number of specific areas. The best known of these are analogue to digital and digital to

analogue conversion, where it is used to obtain high resolution, low bandwidth performance

from a low resolution, high bandwidth converter. A more recent application is CD mastering

from high-resolution recordings stored on digital tape or hard disc. To convert a recording of,

say, 20 or 24 bit resolution to the CD format requires a re-quantization to 16 bits. This inevitably

introduces noise into the signal, and in a straightforward quantization this noise is approximately

white. Noise-shaping can be used create a non-white noise spectrum. For example it is possible

to lower the noise power spectral density in the frequency bands where the ear is most sensitive,

at the expense of higher noise power in other bands (where the ear is less sensitive). This process

lowers the perceived quantization noise or and increases the subjective dynamic range of the

signal.

B. Image Processing

Based on the study written by Bruno A. Olshausen, filtering have been widely used in

digital image processing. The easiest way to smooth a signal is to blur it. In film this is easily

done by adjusting the lens so the image is a bit out of focus. The more out of focus it is, the

lower the spatial-frequency content of the image becomes. And as the spatial- frequency content

is lowered, the temporal frequency content is also lowered. Blurring is often used by film

directors to eliminate aliasing from a scene. For example, when filming a shot where a car is

racing by, it would be a mistake to keep the car in focus. The image of the car is usually

purposefully de-focused so as to eliminate the jerky appearance of motion, which breaks the

illusion of reality in film. Oftentimes though, directors will not bother to eliminate aliasing in the

form of backwards moving wheels, or they may even prefer to keep it for visual effect."

Blurring may be easily accomplished on digital images by filtering (or convolving) with a

mask that averages together the values of several neighboring pixels. Such an operation is

usually provided in image manipulation programs such as Photoshop.

Synthesis of Related Literature to the Case Study

There are many different and unique approaches in designing an Lth Band or Nyquist

Filters. Most of them can be found in books, journals, reports etc. For this project, the

researchers focuses on one single approach that is said to be one of the most useful approach in

designing a filter.

We have mention that 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 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. And the reason for this significance is the use of zero

intersymbol interference property where in the passband gain is normalized to unity.

We, the researchers adopted the approach where in the zero intersymbol interference

property is used.

Digital Lth Band Filter is one useful filter for multi-rate signal processing. Multi-rate

signal processing techniques are widely used in many areas of modern engineering such as

communications, image processing, digital audio, and multimedia. For this project, the

researchers will apply audio as input in our own designed Lth Band filter.