digital signal processing

41 Pages·2017·2.32 MB·English

JANUARY 9, 2017

DIGITAL SIGNAL PROCESSING
Analytical Principles of DSP and Digital Filter Design
PRESENTED TO JAGJIT SEHRA
BY ZEESHAN MUSTAFA LATIF ANSARI
BEng (Hons) Electronics Engineering

ID: s09466807
Contents Page Numbers
Introduction…………………………………………………………………………………………………3
Background………………………………………………………………………………………………….3
Part a – Analytical Problems…………………………………………………………………………7

Question 1, Part (i)………………………………………………………..……………………………….…………7
Question 1, Part (ii)…………..………………………………………………………………………………………8
Question 1, Part (iii)………………………….…………………………………………………………………….10
Question 1, Part (iv)………………………………………………………………………………………………..13
Proof of results obtained in Q1 (ii) by using a suitable m-file………………….…..13
Proof of results obtained in Q1 (iii) by using a suitable m-file………………..……20
Question 2, Part (i)………………………………………………………………………………………………….22
Question 2, Part (ii)…………………………………………………………………………………………………23
Question 2, Part (iii)………………………………………………………………………………………………..24
Question 3, Part (i)………………………………………………………………………………………………….26
Question 3, Part (ii)…………………………………………………………………………………………………27
Question 3, Part (iii)………………………………………………………………………………………………..28
Part b – Digital Filter Design using MATLAB and SPTOOL…………………………….29

Question 4, Part (i)………………………………………………………………………………………………….29
Question 4, Part (ii)…………………………………………………………………………………………………32
Designing LOW-PASS FIR Filter for Noisy Signal…………………………………..……….33
Designing LOW-PASS IIR Filter for Noisy Signal……………………………………….…..35
Question 4, Part (iii)………………………………………………………………………………………………..36
Conclusion………………………………………………………………………………………………….39
References………………………………………………………………………………………………….40
Bibliography……………………………………………………………………………………………….41

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Analytical Principles of DSP and Digital Filter Design

Introduction

The world of science and engineering is filled with signals. DSP refers to various
techniques for improving the accuracy and reliability of digital communications.
Signal processing is the science of understanding the formation of information
conveyed as a function of time, space or any other variable. Analysing this
information requires the acquisition, storage, transmission and transformation of
signals. Signals can be classified as continuous or analogue, such as speech or
discrete, in the form of digital information. Basically DSP works by clarifying, or
standardizing, the levels or states of a digital signal. It is the process of analysing or
modifying a signal to optimize or improve its efficiency or performance (Sehra, 2017).
As per the title states the assessment is undertaken to analyse and observe the
functionalities and complex behaviours of FIR and IIR filters using Matlab software.
In order to complete this assignment, detailed fundamental knowledge of DSP was
acquired including its mathematical techniques for solving equations to design the
digital filters. Based on the knowledge and skills acquired in Signals and Systems
and further in this (DSP) module work was initiated with solving and verifying the
equations given in assessment brief. Following which the required data and results
obtained were implemented into Matlab to analyse, observe and design the digital
filters. Lastly, theoretical values and results calculated were compared with the
values and results which were obtained in Matlab by using suitable program code
and relevant software skills.
Assessment is comprised of two parts; Part A being the “Analytical problems” had to
be solved using various mathematical techniques, learnt throughout the module,
then they were plotted and analysed in Matlab. Part B being the “Digital Filter Design
using MATLAB and SPTOOL” involved designing and filtering the signals. This
involved the design of a noisy guitar signal and development of code (m-file) to
reduce or totally remove the un-wanted noise from it to produce a clean signal which
was also verified by listening both the signals. This section also specifies the design
requirement if a real-time system was to be employed. It ends with discussion on two
current DSP developments kits (available in UK) for audio processing.

Background

In the world of communications circuits it is true whether the signals are analogue or
digital they always contain some level of noise. Noise is said to be the eternal bane
of communications engineers who are always striving to find new ways to improve
signal-to-noise ratio. For this assessment it‟s important to explain the difference
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between continuous-time, discrete-time and digital signals so that further work can
be started (WhatIs.com, 2017).
A continuous signal or a continuous-time signal is a varying quantity (a signal)
whose domain, which is often time, is a continuum (e.g., a connected interval of the
reals). That is, the function's domain is an uncountable set. The function itself need
not be continuous. On the other hand, a discrete time signal has a countable
domain, like the natural numbers.
A signal of continuous amplitude and time is known as a continuous-time signal or
an analogue signal. This (a signal) will have some value at every instant of time. The
electrical signals derived in proportion with the physical quantities such as
temperature, pressure, sound etc. are generally continuous signals. Other examples
of continuous signals are sine wave, cosine wave, triangular wave etc.
Figure 1 shows a typical continuous time signal, function ( ) for the word “signal”.
This is a mathematical representation of the signal as a function of time, t, which is
refered to as the independent variable.

Figure 1 continuous time signal, taken from DSP Moodle
Discrete-time signals can be considered, as digital signals, where the value of the
signal is known at discrete time intervals. All that is known about the signal is a
sequence of values. Both the amplitudes and time can be quantized. In other words,
the digital value can only take a specified value with a specific range. This will
depend on the quality of the conversion process of the A/D.
A digital signal refers to an electrical signal that is converted into a pattern of bits.
Unlike an analogue signal, which is a continuous signal that contains time-varying
quantities, a digital signal has a discrete value at each sampling point. Digital signals
are discrete time signals generated by digital modulation, computers, CDs, DVDs,
and other digital electronic devices (Sehra, 2017).

Difference between analogue and digital signal processing
The basic difference between analogue and digital signal processing is that
analogue signal is a continuous signal which represents physical measurements
while digital signals are discrete time signals generated by digital modulation. Analog
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waves are smooth and continuous while digital waves are stepping, square and
discrete. We can have a model of a system, in the context of signal processing.
Consider the block diagrams shown in figure 2.

Figure 2 a) Analogue recording system b) Digital recording system, taken from DSP Moodle
Discrete-time signals can be considered, as digital signals, where the value of the
signal is known at discrete time intervals. All that is known about the signal is a
sequence of values. Both the amplitudes and time can be quantized. In other words,
the digital value can only take a specified value with a specific range. This will
depend on the quality of the conversion process of the A/D. Figure 3 shows simply
the process of A/D and D/A.

Figure 3 A/D process, taken from DSP Moodle
It should be pointed out that in order to process analogue signals they must be
processed first, that is consider a physical quantity, such as sound signal, it must be
first captured using some form of transducer or sensor before it can be processed. It
must be conditioned, to remove any un-wanted signal information, through filtering.
Once this is complete the analogue to digital process can be done, and the signal
can be manipulated, before it is re-converted back to an analogue form, using an
anti-aliasing filter, sometimes referred as a low pass filter, to remove the stair case
effect of the DAC. Figure 4 below shows a block diagram of an ideal system for
processing continuous time signal (Sehra, 2017).
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Figure 4 Block diagram for processing continuous time signal, Taken from DSP Moodle

The Advantages of DSP
1. It‟s cost effective. Many development kits are available at low cost as well as
of industrial standard.
2. DSP systems can be designed and tested in a simulation using software
tools, which in many cases can guarantee accuracy of systems.
3. Complex processing of information which is difficult or sometimes impossible
in the analogue domain can be accomplished by digital signal processing.
Examples are adaptive filtering, where the digital filter can vary or adapt its
performance to the characteristics of the signal; another is speech analysis
and processing.
4. It‟s reliable and compact.
5. DSP systems will work the same, performance can be enhanced depending
on the evolution of the hardware/software.
6. Multi-processing of information is achieved by DSP.
Disadvantages of DSP
1. DSP is software driven process, up-dates to systems can only be done by re-
programming for example software driven radio.
2. System performance can be reduced. Due to an analogue to digital
conversion, there are bandwidth requirements, which can lead to loss or
quality of information
3. For „real-time‟ applications, it requires a good understanding of mathematical
techniques to fully implement systems to chip (Sehra, 2017).

Applications of DSP
Real time applications of DSP are Audio and multimedia, speech processing and
recognition, image and video processing, Telecommunications, Computer systems
and military systems.
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Part a - Analytical Problems

Causal systems are those that can be implemented in real hardware also called
realisable systems. A “Function” can be a linear but it cannot necessarily be a linear
system. To be a linear system it must be conforming to homogeneity and
superposition. Homogeneity is when the input is multiplied by a constant; the output
is also multiplied by the same constant. Superposition is when the input sequences
of two functions are added, their outputs also are added. A time invariant system is a
system whose output does not depend explicitly on time, it‟s independent of time
(Sehra, 2016).

Question 1, Part (i):

A causal Linear Time Invariant (LTI) System is given by the function H (Z) below:

( )

( )( )

( ) ( ) ( )

Determining the Ideal Nyquist Rate of the above input signal (𝐹 ):
For Ω = 40π
As Ω = 2πf
40π = 2πf

⇒ 20 Hz

For Ω = 80π
As Ω = 2πf
80π = 2πf

⇒ 40 Hz
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Nyquist frequency is twice the maximum component frequency of the function being
sampled.
So Nyquist rate = 2 x 40 = 80 Hz
In other words, in order to retain data it is always sampled at twice the highest
frequency.
( ) ( ) ( )

( ) ( ) ( )

( ) ( ) + 9cos (πn)

As found above = 0, = and = π

Question 1, Part (ii):

Transfer Function Y (Z) = ∑ [ ]

H (Z) = where is Unit delay.
( )( )

=

=
( ( ))

=
( )

=
( )

=

So frequency response ( ) = ∑ [ ] which is Gain of System.

H ( ) =

=

Where = Euler‟s Method = 𝜃 ± 𝑗 𝑖 𝜃
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As = cosө jsinө

When ω = 0
Using Euler‟s Method

H (0) =

( ) ( ( )( )) ( ( )( ))

=

( ( )) ( ( ))

=

H ( ) = = 5.88 ⇒ approximately 6.

When ω =

Using Euler‟s Method

H ( ) =
( ( ) ( )) ( ( )( ) ( )( ))

=

( ) ( )

Here, the identity used is 𝑗

= ⇒

= ⇒ ⇒
( )

=

Here, the identity used is 𝑗

Then we use √ 𝑗 = a + jb which is the magnitude formula in order to get rid of
the complex number.

Therefore, H ( ) = √ 𝑗 ⇒ ⇒ 3.394
√

H ( ) = 3.394

Then in order to find the phase, the formula Ө = is applied, hence;

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Ө = ⇒ ⇒

When ω = π
Using Euler‟s Method

H (ω) =

=

( ) ( ( ) ( ) ( ) ( ))

=

( ) ( )

= ⇒ ⇒ 4
( ) ( )
H ( ) = 4

To determine the ( ) output, all the values of “Gain” and “Phase angles” are
applied to the original equation given below:-
[ ] H ( ) [ ]

[ ] = 10 + ( ) ( )

Calculated “Gains” are multiplied to the “Gains” of the system and then “Phase
differences” are found by subtracting the found phases from the relevant system
phases as shown below:

[ ] = 6 x 10 + 3.394 x 5cos( ) + 4x9( )

[ ] = 60 + 16.97cos( ) + 36( )

For ω = 0 and ω = π, there are no phase angles so there is no phase difference.

Question 1, Part (iii):

To design the filters, first we need poles and zeroes:-

H (Z) =
( )( )

1 = 0

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