week 1 ee518 winter 2014

EE518 PMP Digital Signal Processing Winter 2014 Week 1, January 7

Prof. Les Atlas, [email protected] http://www.ee.washington.edu/class/pmp518/2014wtr/ TA: De Dennis Meng, [email protected] Page 1 of 19

The EE 518 Course See course syllabus and web page (http://www.ee.washington.edu/class/pmp518/2014wtr/) for details, including weekly reading assignments, which are subject to change later in the quarter. Homework #1, due Tuesday 1/14, at 8:50 (beginning of discussion section): From our required class textbook: Oppenheim & Schafer, 3rd Edition, published 2010:

Probs. 2.24, 2.66, 2.73 Homework #1 MATLAB Projects and Exercises (on class web site, along with needed

dtft.m file). Should you be in this class? If your discrete-time signal processing (DSP) and/or mathematics background is weak, please see background material, such as presented in Fall 2013 in EE 505 PMP and/or make use of inexpensive books such as Hayes, Schaums Outline of Digital Signal Processing, Second Edition, 2011 and, if also needed, Hsu, Schaums Outline of Signals and Systems, Second Edition, 2010. This course presumes a solid understanding of linear time-invariant systems, discrete-time signals, basic sampling, Fourier transforms and some background in bilateral z-transforms. You also should have some initial experience in MATLAB.

Review: What you are Expected to Know Oppenheim & Schafer text sections, which you will review this week, seen in Homework #1: Chapter 1. Introduction. Chapter 2. Discrete-Time Signals and Systems. Introduction. Discrete-time Signals: Sequences. Discrete-time Systems. Linear Time-Invariant Systems. Properties of Linear Time-Invariant Systems. Linear Constant-Coefficient Difference Equations. Frequency-Domain Representation of Discrete-Time Signals and Systems. Representation of Sequence by Fourier Transforms. Symmetry Properties of the Fourier Transform. Fourier Transform Theorems. Summary. (Section 2.10 material is not covered in EE 518, yet we use a small amount of this sections random process concepts in a few weeks.) Partially, if theres time: Chapter 3. The z-Transform. Introduction. The z-Transform. Properties of the Region of Convergence for the z-Transform. The Inverse z-Transform. z-Transform Properties and LTI Systems. Summary. A Few (of the very many) Applications of DSP Military

1. sonar and radar and battlefield acoustics Scientific

1. seismic data (first industry to go digital) 2. signal detection (SETI, subatomic physics)

Health 1. medical ultrasound and computed tomography (CT) scans 2. genomics and proteomics informatics

Industrial 1. monitoring manufacturing processes via acoustic and vibration emissions

Commercial/Entertainment and Information Technology 1. CD, DVD, and Bluray players, MPEG, MP3, music, sound, and video compression.



2. Speech recognition (Windows 7 and 8, Apples Siri, Googles Android Voice Search.)

Communications and Information Technology 1. Draft-N, 802ac, and other methods for gigabit wi-fi 2. cell phones 3-4G; such as WiMAX and LTE standards)

Example in figure below, a somewhat dated (e.g. no HDMI interfaces) generic LCD TV architecture,1 where the only non discrete-time portion of this block diagram is the green Analog FE (front end) and the output DACs (digital-to-analog converters).

History of DSP Back in 1960s: * numerical analysis (finite difference), too difficult to solve differential equation in continuous

variables, used discrete simulation * slow simulations of expensive analog systems * speech processing Earlier limitations, such as expensive storage and slow computation, have all come around, and abundant storage and super-fast computation have become advantages of DSP. DSP systems are now easy to design and test:

1. MATLAB and lots of other tools to facilitate digital system design 2. lots of existing code (e.g., FFTW, Fastest Fourier Transform in the West

www.fftw.org) 3. relatively easy to implement new systems in C/C++ or Java or even FORTRAN

Journals for DSP (see www.ieee.org web page)

IEEE Transactions on Signal Processing IEEE Signal Processing Letters IEEE Transactions Audio, Speech, and Language Processing IEEE Signal Processing Magazine (best start for tutorial-type articles)

The above journals are free if access online from University of Washington campus through ieeexplore.ieee.org 1 from Ashwini Raman, Designing with Multi-PLL and Spread-Spectrum Clocks in Digital Entertainment Equipment, Cypress Semiconductor, http://www.cypress.com/?docID=9263 .



Our Notation for Discrete Sequences { }x is a set of indexed continuous-level values { [ ]}x n n= < < , n = integer, which often arise from periodic sampling of analog signal

1[ ] ( ) period frequencyax n x nT T ff

= = = , =

Ex: [ ]x n

Figure 1.1. [1]x =? , [2]x =? , [1.5]x =??

Discrete Time Systems Discrete Time Systems. A discrete time system maps the input sequence, [ ]x n , to a new sequence, the output sequence [ ]y n .

Figure 1.2. Discrete Time System

Note: For the .pdf lecture file which is made after hand-written lecture annotations, the mirror-image of the above figure and most future lecture notes figures might unfortunately sometimes be seen. This is due to a bug which remains in all versions of Microsoft Jounal and/or Adobe Acrobat. It is unavoidable, yet you can always see the correct figure orientation by going back to the original un-annotated EE 518 lecture notes. Definition 1.1. Linear Systems, i.e., Superposition

{}T is linear iff (if and only if): when 1 1{ }y T x= and 2 2{ }y T x= , then: 1 2 1 2 1 2(additive property) { } { } { }T x x T x T x y y+ = + = + and 1 1(scaling property) { } { }T ax aT x= resulting in 1 2 1 2(superposition) { } { } { }T ax bx aT x bT x+ = +



Definition 1.2. Time Invariance or Shift Invariance Shifting the input produces an output that is equivalently shifted, i.e., if [ ] { [ ]}y n T x n= then { [ ]} [ ]T x n N y n N = . Definition 1.3. Causality A system does not respond until after the input starts, i.e., if [ ] 0x n = for 0n n< then [ ] 0y n = for 0n n< . (Output cannot anticipate input.) Definition 1.4. Bounded Input/Bounded Output (BIBO) or Stable If [ ] xx n B n| | < , , then

[ ] yy n B n| | < , . (Wheremeans for all.)

A Brief Note on Convolution There is a nice figure (Figure 2.10) in O&S. Make sure you understand convolution. Two equivalent ways to view convolution

[ ] [ ] [ ] [ ] [ ]k

y n x n h n x k h n k

=

= = are shown in figures 1.3 and 1.4 below.

1. As a combination by sum of sequences, each a weighted and shifted version of the impulse response, shown in Figure 1.3.

Figure 1.3. Convolution: first view.



2. As a set of computations to compute each value of [ ]y n , shown in Figure 1.4

Figure 1.4. Convolution: second view.

Linear time-invariant systems are completely described via convolution with an impulse response [ ]h n . But an often more illuminating way to to characterize LTI systems is via their frequency response. But why?? Frequency Domain Representation of Linear Time-Invariant (LTI) Discrete-Time Systems Consider [ ] j nx n e = as an input, to an LTI system, namely

( )[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]j n k j n j kk k k k

y n x n h n x k h n k x n k h k h k e e h k e

= = = =

= = = = = Then define

( ) [ ]j j kk

H e h k e

=

=

Aside: is an independent variable representing frequency. This ( )jH e notation will make more sense when we see the periodicity section on the next page, and even more sense when we do the z-transform. For the input [ ] j nx n e = , the output [ ]y n is, with our definition above, [ ] ( )j j ny n H e e = Looking at this result in a figure helps show its profound impact:

Figure 1.5. LTI system response (output) to the input j ne .

LTI System [ ]h n

Input [ ] j nx n e =

Output [ ] ( )j n jy n e H e =

Input sequence appears at output with only a change in amplitude and phase.



Thus j ne is an eigenfunction2 for an LTI system. Namely, applying an LTI system {}T to certain special inputs produces outputs that are identical to those inputs multiplied by a (complex) constant. { } [ ] ( ) ( )j n j n j j j nT e y n e H e H e e = = = where ( )jH e is eigenvalue and j ne is eigenfunction. What is this complex constant? ( )jH e is the frequency response of the system, It signifies how the system responds to a particular input frequency. Why is the above important? Most signals can be represented as an infinite sum of complex exponentials. With linearity and time invariance, weighted sums of the above elementary eigenfunctions can exactly predict the LTI system output for these signals.

Periodicity of Frequency Response The frequency response of an LTI system with impulse response [ ]h n is

( ) [ ]j j kk

H e h k e

=

= we can see that

( 2 ) ( 2 )

2

2

( ) [ ] where is any integer

[ ]

( ) ( 1)

j r j r k

k

j k jr k

k

j jr k

H e h k e r

h k e e

H e e

+ +

=

=

=

=

= =

Thus ( )jH e is periodic with period 2 . This periodicity is one reason for the notation used for frequency response in our text.

Representations of Sequences by the (Discrete-Time) Fourier Transform Lets extend the above concept from an impulse response to an arbitrary sequence [ ]x n

( ) [ ] (DT)Fourier Transform or analysis equationj j nn

X e x n e

=

= Is this transform invertible? Yes. As long as [ ]x n can be defined as an infinite superposition of

weighted complex sinusoids of the form 1 ( )2

j j nX e e

.

Since frequency is a continuous variable, this superposition is an integral

1[ ] ( ) Inverse (DT)Fourier Transform or synthesis equation

2j j nx n X e e d

=

What we sometimes just call the Fourier transform here is often called the discrete-time Fourier transform (DTFT) elsewhere.

2 Eigenvectors and eigenvalues in Linear Algebra: let A be a square matrix, x be a vector, be a scalar. If =Ax x for some x and , then x is eigenvector and is eigenvalue for matrix A . Applying the system of matrix multiplication to x is equivalent to multiplying x by a constant.



( )( ) ( )jj j j X eX e X e e = where ( )jX e | | is the signals magnitude frequency response and

( )jX e is the signals phase response or phase shift. Note: Phase, as so far defined, is not unique. For example, ( ) ( ) 2j jX e X e k = + , where k is any integer. Comparison:

( ) [ ] (Discrete Time) Fourier Transformj j nn

X e x n e

=

=

Relates to upcoming z-transform via jz e (The unit circle of complex domain z, in polar form , 1jz z e z= = .)

( ) ( ) (Continuous Time) Fourier Transformj tX j x t e dt

=

Relates to the Laplace transform via s j j= + (The imaginary axis of complex domain s, in rectangular form Im{ } Im{ }s j j= + = .) Both have units of frequency, but with considerable differences, which we will be carefully facing:

For discrete-time sequences the frequency variable is radians (dimensionless) For continuous-time signals the frequency variable is radians/second

Some Symmetry Properties of Fourier Transform Defining this compact notation for a Fourier transform/inverse transform pair [ ] ( )F jx n X e then [ ] ( )F jx n X e where * means complex conjugate.

Similarly it can be shown that [ ] ( )F jx n X e

For the remaining symmetry properties, it is helpful to define the following sequences:

( )

( )

[ ] [ ] [ ] [any sequence can be expressed as a sum of symmetric sequences]

1[ ] [ ] [ ] [ ] [conjugate-symmetric (or even) sequence (if [ ] is real)]21[ ] [ ] [ ] [ ] [conjuga2

e o

e e

o o

x n x n x n

x n x n x n x n x n

x n x n x n x n

= +

= + =

= = te-antisymmetric (or odd) sequence (if [ ] is real)]x n

Similarly,

( ) ( ) ( ) [A FT can be expressed as a sum of symmetric functions]

1( ) ( ) ( ) ( ) [conjugate-symmetric (or even) function (if ( ) real)]21( ) ( ) ( )2

j j je o

j j j j je e

j j jo

X e X e X e

X e X e X e X e X e

X e X e X e

= +

= + =

= ( ) [conjugate-antisymmetric (or odd) function (if ( ) real)]j j

oX e X e =



Given these definitions, we can now derive the FT transform pair

Re( [ ]) ( )F jex n X e

as follows

{ } ( ){ } { } { }( )1 1 12 2 2Re( [ ]) [ ] [ ] [ ] [ ] ( ) ( ) ( )j j jeFT x n FT x n x n FT x n FT x n X e X e X e = + = + = + = In similar fashion we can obtain

Im{ [ ]} ( )F joj x n X e ,

{ }[ ] ( ) Re ( )F j je Rx n X e X e = , and { }[ ] ( ) Im ( )F j jo Ix n jX e j X e =

Symmetry Properties of the FT for Real Sequences For any real [ ]x n ( ) ( )j jX e X e =

i.e., the FT has conjugate-symmetry for real signals. Both sides of this expression can be expressed in Cartesian form as a sum of its real and imaginary components like this:

( ) ( ) ( ) ( )j j j jR I R IX e jX e X e jX e + =

from which it follows by separating the real and imaginary parts that (for real sequences!) the real part is even, ( ) ( )j jR RX e X e

= and the imaginary part is odd, ( ) ( )j jI IX e X e

= .

Furthermore, by expressing ( )jX e in polar form

( )( ) ( )jj j j X eX e X e e =| |

it can be shown for real sequences that the magnitude is an even function of ( ) ( )j jX e X e | |=| | and the phase is can be chosen to be an odd function of ( ) ( )j jX e X e = . Also, [ ] ( )F je Rx n X e

So, a sequence that is real and even in time has a FT which is real and even in frequency. These properties and a few more are given in O&S Table 2.1 on page 56.



Fourier Transform Theorems Let the FT of [ ]x n and [ ]y n be ( )jX e and ( )jY e , respectively.

Linearity [ ] [ ] ( ) ( )F j jax n by n aX e bY e + +

Delay [ ] ( )dj nF jdx n n e X e

Parsevals Theorem

2 21[ ] ( )

2j

nx n X e d

=

| | = | |

Convolution (important, so we spend more time on it) [ ] [ ] ( ) ( )F j jx n y n X e Y e Convolution in time corresponds to multiplication in frequency. Time Reversal If [ ] ( )F jx n X e , then

[ ] ( )F jx n X e If [ ]x n is real for all n , then ( ) ( )j jX e X e = . It then follows that [ ] ( )F jx n X e Derivative in Frequency If [ ] ( )F jx n X e , then

( )[ ]

jF dX enx n j

d

Modulation (Windowing) Theorem If [ ] ( )F jx n X e and [ ] ( )F jw n W e then

( )1[ ] [ ] [ ] ( ) ( )

2F j jy n x n w n X e W e d

=

So pointwise multiplication in time corresponds to periodic convolution in frequency.

What if we want to know more than a systems steady-state response or its steady-state output for a steady-state input?



Chapter 3. The z-Transform The Fourier transform of [ ]x n is

( ) [ ]j j nn

X e x n e

=

=

Definition 1. The bilateral z-transform is

( ) [ ] nn

X z x n z

=

= where z is a complex continuous variable.

Compact notation [ ] ( )Zx n X z Note: Consider

( ) ( ) [ ]j j j nz en

X z X e x n e

=

=

| = =

when the sum exists for the values of z where 1z| |= . In general jz re =

So

( ) ( ) [ ]( ) ( [ ] ) { [ ] }j j n n j n nn n

X z X re x n re x n r e FT x n r

= =

= = = =

( )X z is complex function of a complex variable defined on the z-plane, shown in Fig 1.6.

Note ( )jX e is evaluation of z-transform around the unit circle. 0 1( ) ( )j

zX e X z

= =| = | ,

2( ) ( )j z jX e X z

==| = | . So the Fourier transform is unwrapping of the unit circle periodically into

the line.

Figure 1.6. The complex z-plane

{ }Re z

{ }Im z



Convergence of the z-Transform ( )X z might not converge for all z . For uniform convergence, absolute summability requires

[ ] nn

x n z

=

| |< The z-transform might converge, for some values of z, even if the Fourier transform (FT) does not. Definition 2. Region of Convergence or ROC

{ [ ] }nn

ROC z x n z

=

: | || | < i.e., locations in z of absolute summability. So, if 1 ROCz , then 1 ROC

jz e | | for all . (the ROC boundary depends only on the magnitude of z.) For the signals we work with, the ROC can be a disk or a ring (donut) in the z -plane, as shown in Fig 1.7.

Figure 1.7. Region of Convergence

The FT exists if and only if the ROC contains the unit circle 1z| |= . Uniform convergence requires absolute summability of [ ] nx n z . Note: we normally think of the FT as evaluation of z-transform at 1z| |= . Definition 3. Rational z-Transform A z-transform, when it can be summed and expressed in the simple form:

( )( )( )

P zX zQ z

=

is called rational z-transform, where ( )P z and ( )Q z are polynomials in z . The zeros of ( )X z are { ( ) 0}z X z: = The poles of ( )X z are { ( ) }z X z: =



Definition 4. Right Sided Sequence [ ]x n is right sided if there exists an N such that [ ] 0x n = for n N< < . Questions: Is causal sequence right sided? Yes. Is right sided sequence causal? No, not necessarily, since N could be less than 0. Definition 5. Left Sided Sequence [ ]x n is left sided if there exists an N such that [ ] 0x n = for n N> > . Definition 6. Two-Sided Sequence If [ ]x n is infinitely long but neither left sided nor right sided, then [ ]x n is two-sided. Ex 1: [ ] [ ]nx n a u n= . Note: a can be complex n n j na a e =| |

1

0 0( ) ( )n n n

n nX z a z az

= =

= = Convergence requires

1

0

n

naz

=

| | < or z a| |>| | So

11( )

1zX z z a

az z a= = | |>| |

There is a pole at z a= and a zero at 0z = . Ex 2: [ ] [ 1]nx n a u n= which is a left sided sequence.

1

1

1 0( ) 1 ( )n n k k k

n k k

zX z a z a z a zz a

= = =

= = = =

Convergence requires 1 1a z| |< , i.e., z a| |



Properties of the Region of Convergence (ROC) for the z-Transform Property 1. The ROC is a ring or a dish in z-plane centered at the origin, i.e., 0 R Lr z r

Note: 0[cos( )] [ ]n u n is a right-sided sequences whose z-transform converges for 1z| |> , yet its Fourier transform doesnt exist in strict uniform convergence sense. (See O&S Table 3.1)



Figure 1.8. A 3-dimensional depiction of a z-transform magnitude in dB:

1 2

1 21 2 4 2 881 0 8 0 64

20 log ( ) 20log dBz zz z

H z . + . . + .= . It has poles around 0 4 0 6928j. . and zeros around 1 2 1 2j. .

Why do we care about z-transforms?? One main reason

[ ] [ ] [ ] ( ) ( ) ( )Zy n x n h n Y z X z H z= = So as in the case of Fourier transform, z-transforms techniques simplify the mathematics needed to characterize and solve for the responses of LTI systems. But the steady-state restriction of Fourier transform solutions are greatly expanded via more general solutions of the z-transform. But if we can now find an ( )X z , ( )H z , and ( )Y z , how can we then determine the corresponding [ ]x n , [ ]h n , and/or [ ]y n ? This next section covers that.



The Inverse z-Transform The inverse z-transform equation can be written as

11[ ] ( )

2n

Cx n X z z dz

j=

Integrating the real variable over ( , ) is equivalent to integrating the related complex variable jz re = over the contour C, which is denoted by the contour integral .

C The contour C is a counter-clockwise closed circular contour centered at the origin with radius r such that { } ROCz z r: | |= ,

because r may be anything in the region of convergence. Easier Ways to Compute the Inverse z-Transform Inspection Technique Use the known z-transform pairs (O&S table 3.1, page 110) and the z-transform properties (such as linearity!) (O&S table 3.2, page 132). Partial Fraction Expansion Technique There are some common forms of the z-transform for which we can get the inverse relatively easily by inspection. The goal of the partial fraction expansion technique is to get a given z-transform expression in such a form. When

0

0

( )( )( )

M kkk

N kkk

b zP zX zQ z a z

=

=

= =

We can factor the denominator and numerator and get

1

0 11

0 1

(1 )( )

(1 )

Mkk

Nkk

c zbX za d z

=

=

=

where ck represents the non-zero zeros and dk represents the non-zero poles of X(z). Case 1. N M> and the poles are all first order poles (i.e., all the poles are unique).

11

( )1

Nk

k k

AX zd z=

=

Note: The residue for the pole kd is 1(1 ) ( )

kk k z d

A d z X z=

=

since

1

11

1

(1 )(1 ) ( )1

Nl k

k kl k l l

A d zd z X z Ad z

, =

= +

The first term is obtainable by long division (example later). The second term is obtainable by the same procedure used when .N M> This technique of finding the partial fraction expansion of a z-transform is referred to as the Heaviside cover-up method, and works also for cases with repeated roots (see text and links on class web site for more details). In its most general form, the z-transform can be written as



1max( 0)0

10

0

max( 0)

10 1 1

( ) ( )

1

u i

M kM Nkr k

r N kr kk

N SM Nr ik

rr i k i

b zX z B z M N

a z

AB zd z

, =

=

=

,

= = =

= +

week 1 ee518 winter 2014

Documents