ECE 524 notes 4 - radio frequency circuit design PDF

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ECE 524E – DIGITAL SIGNAL PROCESSING

(Mr. Chemweno)

ECE 524E: DIGITAL SIGNAL PROCESSING Course outline Semester/Year/Lecturer Sem I, Year 5, Mr. Chemweno E. K Units 3 Pre-Requisites: ECE 433 Control Systems II, ECE 321 Signals, ECE 438, Digital Control Aims/Objectives/Purposes  To introduce the design concepts and realization of Finite Impulse and Infinite Impulse Response types of Digital filters.  To introduce the student to frequency domain analysis of discrete-time signals  The student will also learn hardware and software design and implementation. Learning outcomes At the end of the course the student should be able to:  Characterize signals and sytems using Z-transforms and inverse Z- transforms.  Design and realize Finite Impulse Response (FIR) and Infinite Impulse Respons (IIR) filters to satisfy given specifications.  Define and apply Fourier transforms in spectral analysis and design (DFT and FFT).  Carry out a mini project in the design of filters Content Digital filter design: Review of discretization of signals and the z-transform; Design of Finite Impulse Response (FIR) filters, their properties and applications. Windowing. Design of Infinite Impulse Response (IIR) filters, their properties and applications. Realization of FIR and IIR filters. Discretization of analog filters. Computer-aided filter design. Frequency domain analysis of discrete-time signals: Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT) and their properties. Application of Fourier transforms in spectral analysis and design. Hardware and software design and implementation: Case studies in areas such as audio and video signal processing and remote sensing; implementation using DSP chips, Microcontrollers and Software realization. Mini-project. Teaching/delivery methods Lectures, Tutorials, Labs, Group Discussions, Demonstration, Special Projects Assessment Examination 70%, Continuous Assessment Tests (CATs) 15%, Assignment 10% Labs 5%, Total 100%

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ECE 524E – DIGITAL SIGNAL PROCESSING

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References 1. 2. 3. 4. 5. 6.

Introduction to digital signal processing – Roman Kuc Digital signal processing – Openheim and Schaffer Fundamentals of digital signal processing – Lonnoie C. Luderman Digital signal processing – John G. Proakis and Dimitris Manolakis Digital signal processing - A computer based approach –Sanjit K. Mitra The scientist and Engineers guide to Digital signal processing –Steven W. Smith

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ECE 524E – DIGITAL SIGNAL PROCESSING

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1.0 INTRODUCTION A signal is a function of one or more independent variable(s) such as time, position, temperature etc. For example, speech and music signals represent the variations of air pressure as a function of time at some point in space; while a picture is represented as the intensity of light in a two dimensional spatial coordinates. Signals generally convey information about the state or behavior of a system. The objective of signal processing thus is to extract useful information carried by the signal. Most signals encountered in real life are generated by natural means such as speech, vibrations etc. Further, a signal may be generated synthetically or by computer simulations. Signal processing may involve processes such as filtering to remove unwanted noise and interference, equalization, etc. The method of extracting the information depends on the type of the signal and the nature of the information being carried by the signal. In the foregoing discussions, we shall treat the independent variable as time, though they may not be in actual sense. Most signals encountered in life are analogue in nature and may be processed directly by analogue systems for the purposes of changing their characteristics or extracting some desired information.

Analogue input signal

Analogue signal processor

Analogue output signal

Figure 1.1 : Analogue signal processing Digital signal processing provides an alternative method for processing analogue signals Analogue input signal Digital signal processor

A/D Converter

D/A Converter

Digital output

Digital input

Figure 1.2 : Digital signal processing of analogue signals

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Analogue output signal

ECE 524E – DIGITAL SIGNAL PROCESSING

(Mr. Chemweno)

(a) Analogue input to be processed

(b) Sample and Hold circuit output

(c.) A/D convertor output (uncoded and coded)

(d) Processor (uncoded and coded)

(e) D/A converter output

(f) processed analogue output

Figure 1.3 : typical signal waveforms appearing at various stages of DSP system

1.1 CHARACTERIZATION AND CLASSIFICATION OF SIGNALS Signals may be classified as: (a) Continuous time vs discrete time (b) Continuous valued vs discrete valued (c) Real valued or complex (d) Random or deterministic (e) One dimensional or multi dimensional

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ECE 524E – DIGITAL SIGNAL PROCESSING

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1.2 Definitions (a) Continuous time signals – Signals defined at a continuum of times and are represented by a continuous variable. Example s (t )  A sin 3t (b) Discrete time signals – Signals defined at discrete times and thus, the independent variable takes only discrete values. Discrete signals may be represented as a sequence of numbers in the following ways: n  1,3 1  x (n )   4 n2 Functional representation 0 elsewhere  Tabular representation n   2 1 0 1 2 3  x(n ) 0 0 0 0 1 4 1 0

x(n)

Sequence representation

 0 0 1 4 1 0  

Note that the  indicates time n  0 (c) Digital signals – Signals whose both amplitude and time are discrete (quantized) (d) Analogue signals – Continuous time and continuous amplitude signals (e) Continuous time system – systems for which both the input and the output are continuous time signals (f) Discrete time system – systems for which both the input and output are discrete time signals. It is important to note that a system not only includes the physical devices, but also the software realizations of the operations on a signal. 1.3 APPLICATION OF DSP Some of the time domain signal processing operations include amplification/attenuation, integration/differentiation of signals, filtering, modulation/demodulation, multiplexing/demultiplexing etc. DSP finds applications iin diverse fields such as military, space, medicine, communication, scientific researches etc. Specifically, some of the applications include: 

Automatic speech recognition 5

ECE 524E – DIGITAL SIGNAL PROCESSING

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   

Speaker identification/verification Speech data compression Signal scrambling Speech synthesis

      

Image reconstruction Image enhancement Image coding Image analysiss Equalizers Filters Echo cancellation

 

Telephone dialing Circuits for special effects such as FM stereo etc.

1.4 ADVANTAGES AND DISADVANTAGES OF DSP 1.4.1 ADVANTAGES (a) Flexibility in configuring DSP operations by simply changing the programs. Analogue systems would imply redesign of the system (b) Provides a much better control of accuracy requirements by specifying the requirements of A/D converter in terms of word length, floating point vs fixed point arithmetic etc (c) Digital data is easily stored in tapes and disks without deterioration or loss of fidelity beyond that introduced by the A/D conversion (d) Cheaper implementation than analogue counterparts since digital hardware is cheaper as a result of flexibility of reprogramming (e) Sharing of a processor by different signals by multiplexing (f) Allows for realization of certain characteristics not possible with analogue implementation such as linear phase (g) They may be cascaded without loading problems (h) Processing of very low frequency signals such as seismic signals whereby capacitors and inductors would be very large in size 1.4.2 DISADVANTAGES (a) Increased system complexity – additional pre and post processing devices (b) Limited range of frequencies available for processing. Sampling theorem requires that the signal be sampled at a frequency atleast least twice the highest frequency component.

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ECE 524E – DIGITAL SIGNAL PROCESSING

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(c) Digital processing systems are constructed from active devices which consume power, while analogue processing devices may require only passive devices

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ECE 524E – DIGITAL SIGNAL PROCESSING

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2.0 DISCRETE TIME SIGNALS AND SYSTEMS Discrete time signals may arise from (a) Sampling of continuous signals (b) Discrete processes such as accumulating a variable over a period of time If a continuous signal xa (t ) is sampled at every T seconds, a sequence x(nT ) results. We shall only consider sequences that are sampled at equal time intervals. For convenience, the sampling reference T will be dropped and x(n) will be used to represent the sequence. Note that the term discrete time may refer to sequences of other physical parameters such as discrete location, discrete distance or discrete position. x a (t )

x(n )

t  nT

t

Sampler

0

1

2

3

4

5

6

7

8

n

Figure 2.1 : Sampling of a continuous time signal x( n)  x a (nT )

  n  

The spacing between two consecutive samples is called the sampling interval or the sampling period. Fs 

1 , sampling frequency T

1.5 SOME IMPORTANT SEQUENCES (a) Unit sample sequence n 0 1  (n )   0 Otherwise A delayed version of the unit sample sequence is denoted nk 1  (n  k )   0 Otherwise

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ECE 524E – DIGITAL SIGNAL PROCESSING



 



-1

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 0





-2

1





2

3

n



  -2

 -1

Impulse function



0



1

 

 k



k+1 



Delayed impulse function

An important property of the unit impulse sequence is that it can be used to select (or sift) one sample x(k ) from a sequence x(n). This is known as the sifting property and expressed as: x (k ) 



  (n  k )x (n )

n  

(b) Unit step sequence n 0 1 u (n )   0 Otherwise 

 



 -1

0

1

2

3

n



Unit step sequence

The unit sample and the unit step sequences are related as follows:  (n )  u (n )  u (n  1) 

u( n)    ( n  k) or u (n )  k0

n

 (k )

k  

The unit step sequence may be used to define the starting point of a sequence in analytic expressions

a n n 0 x (n )    x (n )  a n u (n ); for _ all _ n  0 elswhere (c) Unit ramp sequence n0 n u r (n )   0 Otherwise (d) Real exponential sequence x (n)  a n ; for _ all _ n

(e) Sinusoidal sequence x (n )  cos(o n   ) 9

ECE 524E – DIGITAL SIGNAL PROCESSING

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x (n )  sin( o n   )

The unit step sequence and the real exponential sequence have properties similar to their continuous time counterparts. We however note the following for sinusoidal sequences: (i) A sinusoidal sequence x(n) is said to be periodic with period N if x( n)  x( n  N ) : n If x (n )  A cos(o n   )  A cos(o n  o N   ) A, o , referred to as amplitude, angular frequency and phase respectively

 o N  2 k :   0,1,2,3 2 The period N  must be an integer for the sequence to be periodic o  This can also be stated that o must be a rational number that can be expressed 2 p for the sequence to be periodic q 2 (ii) If is not an integer but a rational number, the sinusoidal sequence will be o 2 periodic, with a period longer than o 2 (iii) If is not rational, then the sequence is not periodic o (iv) Discrete time sequences, whose frequencies are separated by integer multiples of 2 are identical. Consider the sequences of cos(0.1 ) and cos(2.1 )

as

cos(2.1 )

cos(0.1 )

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

-0.2

-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8

-1

-1 0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20

Figure 2.2 : Two analogue signals having frequency difference of integer multiples of 2 have identical sequences. 10

ECE 524E – DIGITAL SIGNAL PROCESSING

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This can easily be shown from the relationship: If  1  o  2 m , m  integer Then cos(1 n)  cos(o n) (v) The set of all distinct values that a discrete time sinusoidal sequence can have occurs for values of  ranging from       or simply [  ,  ] Because of these last two properties, any sinusoid having the frequency    is called an alias.

(f) Complex exponential sequence x (n )  e (  j) n

An arbitrary sequence can be expressed as a sum of scaled and delayed unit samples as: p(n)  a 3 ( n  3)  a1 (n  1)  a 2 (n  2)  a 7 ( n  7) p (n )

a3 a1 

 4

3

 2

 1

0

1

2

 3

 4

 5

 6

7

 8

a7 a2

Figure 2.3: An arbitrary sequence This can be written in general terms as x (n ) 



 x (k ) (n  k )

k  

1.6 LINEAR SHIFT INVARIANT SYSTEM A discrete system may be thought of as a unique transformation or operator that maps some input sequence x(n) to some output sequence y(n)through some transformation T [] such that y (n )  T [x (n )] T []

x(n)

11

y(n)

n

ECE 524E – DIGITAL SIGNAL PROCESSING

(Mr. Chemweno)

Figure 2.4 : Representation of a system A linear system obeys the principle of superposition which states: If y1 ( n)  T[ x1 (n)] and y2 ( n)  T[ x2 (n)] , then a system is linear if

T[ x1 ( n)  x2 ( n)]  T[ x1 ( n)]  T[ x2 (n)]  y1 (n)  y2 (n )  ,  are constants Example 2.1 : Determine the linearity of a 3-sample average given by 1 y( n)  [x (n  1)  x (n )  x (n  1)]  T [x (n )] 3 Solution: T[x1 ( n)  x2 (n)]  

1 [x1 (n  1)  x2 (n  1)  x1 (n )  x2 (n )  x1 (n  1)  x2 (n  1) 3

 3

[ x1 ( n  1)  x1 ( n)  x1 ( n  1)] 

 3

[ x2 ( n  1)  x2 ( n)  x2 ( n  1)]

  y1 ( n)   y2 ( n) Hence linear. Example 2.2 : Determine the linearity of the system y( n)  T [ x( n)]  x 2 ( n)

Solution: T[x1 ( n)  x2 (n)]  [ x1 ( n)   x2 ( n)]2

  2 x12 (n)   2x 22 (n)  2x1 (n) x2 (n)   2 x12 (n )   2x 2 (n) Hence non linear 2

A discrete time system is shift invariant for all n and n o y( n  no )  T [ x( n  no )] , n o is the number of delay samples.

Example 2.3: A system is described by y (n )  T [ x (n )]  nx (n ) Determine whether the system is linear and whether it is time varying Solution 12

ECE 524E – DIGITAL SIGNAL PROCESSING

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Linearity: T[x1 ( n)  x2 (n)]  [ nx1 ( n)   nx2 ( n)]2

 [ nx1 ( n)]  [nx2 (n)]   y1 ( n)   y2 ( n) Hence lnear Shift invariance y( n  no )  T[ x( n  no )]  n  no x( n  no ) Since the coefficient is time varying, then the system is not shift invariant. A system that satisfies both the above two conditions is known as Linear Shift Invariant system.

1.7 UNIT SAMPLE RESPONSE A Linear shift invariant system can be characterized by its unit sample response h(n) . That means that h(n) provides all the information needed to determine the response to any input. Consider y (n )  T [ x (n )]     T   x (k ) (n  k )   k    Using the principle off superposition, we can consider this as delayed unit sample sequences scaled by x(k ) such that y (n ) 



 x (k )T [ (n  k )]

k  

y (n ) 



 x (k )h (n  k )

k  

The above equation is known as the convolution integral, and can be re written y (n ) 





k 

k  

 x (k )h (n  k )   x (n  k )h (k )

By definition y (n )  x (n )  h (n ) Example 2.4: 13

ECE 524E – DIGITAL SIGNAL PROCESSING

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Determine the unit sample response of the 3 – sample average 1 y( n)  [ x (n  1)  x (n )  x (n  1)] 3 Solution: Let x (n )   (n ) , input is unit sample 1 y( n)  [ ( n  1)   ( n)   ( n  1)] 3 1 For n  3, y ( 3)  [ ( 2)   (3)   ( 4)]  0 3 1 For n  2, y ( 2)  [ (1)   (2)   (3)]  0 3 1 1 For n  1, y ( 1)  [ (0)   ( 1)   ( 2)]  3 3 1 1 For n  0, y (0)  [ (1)   (0)  (1)]  3 3 1 1 For n  1, y (1)  [ (2)   (1)   (0)]  3 3 1 For n  2, y ( 2)  [ (3)   (2)   (1)]  0 3

1 n  1,0,1 Thus h (n )   3 else 0 Example 2.5: Determine the unit sample response of a first order recursive filter n 0  ay( n  1)  x( n) y (n )   0 otherwise  Solution: h (n )  y (n ) x ( n) (n )

For n  0, h (n )  0 For n  0, h(0)  ay( 1)   (0)  1 For n  1, h(1)  ay(0)   (1)  a For n  2, h(2)  ay(1)   (2)  a

2

For n  3, h(3)  ay(2)  (3)  a 3 n Hence h( n)  a u(n)

Example 2.6: Unit sample sequence for a time varying system 14

ECE 524E – DIGITAL SIGNAL PROCESSING

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 1  y (n  1)  x (n ) n0 Determine h(n) for the system y( n)   n  1  otherwise 0 Solution At time n  0 h ( 0)  1

h (1, ) 

1 2

h ( 2, )  16 h(3)  241

At time n  1 h (0)  0 h(1, )  1 h( 2, )  13

h(3)  121 Note that the filter response is time varying 1.8 COMPUTATION OF THE CONVOLUTION INTERGRAL y (n )  h (n )  x (n )  x (n ) h (n ) That is commutative property It also obeys the distributive property:

h( n)  x1 ( n)  x2 ( n)  h( n)  x1 ( n)  h(n)  x2 (n) The convolution integral may be determined analytically or graphically. 1.8.1 Analytic evaluation (a) Choose an initial value of n , for which to start the summation. If x(n) sequence starts at time n x and h(n) starts at n h , then n  nx  n h (b) Express both x (n ) and h (n ) as x(k ) and h (k ) . Note that x(k ) resembles x (n ) and h (n  k ) is the time reversed version of h (n ) (c) Multiply the sequences element by element and the products are accumulated over all value...