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600 ECE Lecture Notes, Part #1 Outline, Part #1 Introduction Signals & Systems Review Time Frequency Sampling Aliasing Reconstruction Rate conversion Copyright 2004-2008, L.C. Potter, The Ohio State University ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 1 Signals A signal conveys information transducer voltage aircraft velocity cellular phone downlink music backscattered electric eld phosphorous diffusion rate electrocardiogram speech charge distribution bat chirp photographic image sunspot data stock prices motor rpm Abstractly, a signal is a function of one or more independent variables ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 2 Continuous and Digital Continuous time speech signal 12 Sequence of samples, T=125 milliseconds 12 10 10 8 8 6 6 millivolts millivolts 4 4 2 2 0 0 2 2 4 4 6 0 5 10 15 20 25 30 35 6 0 50 100 150 200 250 milliseconds 257 samples ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 3 Why Digital? Moore's Law exponential growth in electronic processing capability transistor 1947: Bardeen, Brattain & Shockley (1956 Nobel Prize) integrated circuit 1958: Kilby & Noyce Flexibility reprogrammable, re-usable architecture same architecture can implement linear, adaptive, and nonlinear processing allows for linear phase lters and error correcting codes Precision and reliability bits versus tolerance reproducible results; no tweaking robust to aging, temperature, power supply, etc. Cost OHIO S ATE T UNIVERSITY T . H . E ECE600 Part #1 Slide # 4 Why Digital? ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 5 Digital Processing of Continuous-time Signals x(t) analog R,L,C & op amps y(t) 0111 ... x(t) H (s) pf A/D digital filter 0101 ... D/A H r (s) y(t) Example: AT&T IDSN microprocessor TMS32010 (1983) clock rate 5,000,000 operations per second sampling rate 8 kHz Clock and sampling rates imply up to 625 instructions per sample for real-time applications, such as echo cancellation, multiplexing, etc. ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 6 Systems A system is an interconnection of components (devices or processes) with a set of cause and effect relationships human voice wideband radar computed tomography imager ight navigation R-L-C circuit digital communications link anti-lock braking human ear speaker recognition device market economy Abstractly, a system is a set of relationships between input signals and output signals inputs system outputs ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 7 System Example: Communcation Link RF and Deep Space Links on Mars Path nder Mars Sojouner Rover Gladstone Antenna estimate of message message transmitter channel receiver High-level system representation of a communication system mars.jpl.nasa.gov ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 8 System Example: Feedback Control Anti-lock braking system (ABS) disturbance reference + + output controller - plant + sensor(s) Block diagram of a feedback control system www.abs-education.org ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 9 System Examples: Image Processing Computed tomography imaging for non-destructive testing engine turbine blade tree log Jet engine turbine blades are a common casting application for industrial CT. Digital imaging systems provide a wider dynamic range than lm-based systems. The lumber industry has successfully used limited angle CT to triangulate the location of aws in logs. Flaw information is used to determine the most ef cient saw plan that will maximize yield. www.bio-imaging.com ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 10 System Example: Radar Imaging Mathematical system model similar to computed tomography ARL BoomSAR Image of vehicles behind trees www.arl.army.mil/alliances ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 11 System Example: Amortization Amortization Table: Mortgage Schedule of Payments Month 1 2 3 4 . . . 11 12 . . . 357 358 359 360 Payment $599.55 $599.55 $599.55 $599.55 . . . $599.55 $599.55 . . . $599.55 $599.55 $599.55 $600.00 Balance $99,900.45 $99,800.40 $99,699.85 $99,598.80 . . . $98,877 $98,771 . . . $1781.25 $1190.61 $597.01 $0.00 Principal paid $99.55 $100.05 $100.55 $101.05 . . . $104.64 $105.16 . . . $587.71 $590.64 $593.60 $597.01 Interest paid $500.00 $499.50 $499.00 $498.50 . . . $494.91 $494.39 . . . $11.84 $8.91 $5.95 $2.99 Inputs: schedule above computed for a $100,000 loan, 30 year xed, and 6% interest rate. ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 12 System Models For analyzing and describing systems, engineers use models. A mathematical model is a set of rules (often equations) describing the relationships between signals appearing in the system. A model is an idealized representation, so there is a trade-off between simplicity and accuracy of the model. ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 13 System Models All models are false, but some are useful. G. E. P. Box (http://www.asq.org/about/history/box.html) La simplicit c'est la plus grande sagesse. e unknown Seek simplicity and distrust it. Alfred North Whitehead (1861 1947) Our life is frittered away by detail ... simplify, simplify! Henry David Thoreau (1817 1862) Trust the math, but question the assumptions. John McCorkle OHIO S ATE T UNIVERSITY T . H . E ECE600 Part #1 Slide # 14 An Example Linear System Model x[n] b0 y[n] -a1 -a2 -a3 Recursive difference equation: Thus, the output is a scaled sum of past and present inputs and past outputs. To implement a difference equation in digital hardware, only three building-block elements are required: storage register (memory element or delay register, making the lter a dynamical system); through ) is used in the speech scaling; add. This model (with coef cients compression system known as code excited linear prediction (CELP). ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 15 Signal Processing Themes Principles are widely applicable use simple techniques to solve complex problems System and signal modeling represent salient behavior; analysis and design systems can be represented by signals Frequency decomposition of signals Linear time-invariant system models convolution frequency response Optimization of a design criterion: least-squares; min-max OHIO S ATE T UNIVERSITY T . H . E ECE600 Part #1 Slide # 16 Why Consider DSP? Goal in ECE600 is to prepare informed users interpretation of results choice of processing understanding trade-offs Typical application elds wireless communications automatic control medical imaging remote sensing econometrics voice recognition ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 17 Signals & Systems Review Time-domain concepts are rst reviewed, then frequency domain ideas are surveyed. Signals Systems Linearity & Time-invariance Convolution Difference equations Frequency response Sinusoidal steady state response Discrete-time Fourier transform Different representations of linear systems provide perspectives for understanding, designing and debugging: impulse response, step response, frequency response, transfer function, difference equation, block diagram, state-space equations. ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 18 Time Domain Signals A discrete-time signal is a function de ned on the integers; that is, a sequence of numbers. Examples explicit: M ATLAB: >> x = [3 9 1/3 2 -4 pi]; implicit: nite duration ! " % complex-valued: ECE600 Part #1 # $ & ' OHIO S ATE T UNIVERSITY T . H . E Slide # 19 Important Signals: Impulse unit sample sequence (impulse or Kronecker delta ) Useful for decomposition of arbitrary signals and for system modeling. unit impulse sequence 1 0.8 amplitude 0.6 0.4 0.2 0 0.2 5 4 3 2 1 0 1 2 3 4 5 sample number, n >> >> >> >> >> n = [-5:1:5]; %time instants of interest x = zeros(size(n)); x(find(n==0))=1; %all zeros, then 1 at origin stem(n,x,'filled'); axis([-5.1 5.1 -0.2 1.2]);%stem-style plot & adjust axes title('unit impulse sequence','fontsize',20)%make a title xlabel('sample number, n','fontsize',20);ylabel('amplitude','fontsize',20) OHIO S ATE T UNIVERSITY T . H . E ECE600 Part #1 Slide # 20 Important Signals: Unit Step unit step sequence Useful for modeling initiation of a signal, a data collection, or a process. unit step sequence 1 0.8 amplitude 0.6 0.4 0.2 0 0.2 3 2 1 0 1 2 3 4 5 sample number, n >> >> >> >> >> n = [-3:1:5]; %time instants of interest u = zeros(size(n));%start with all zeros list of samples u(find(n>=0))=1; % replace zeros with ones to make step function % using ``find'' command to identify nonneg indices stem(n,u,'filled'); %make the plot in the ``stem'' style OHIO S ATE T UNIVERSITY T . H . E ECE600 Part #1 Slide # 21 Important Signals: Pulse pulse sequence otherwise pulse sequence, L=13 A rectangular window modeling the duration of an event or data collection. 1 0.8 amplitude 0.6 0.4 0.2 0 0.2 5 0 5 10 15 sample number, n >> >> >> >> n = [-8:1:20]; % time instants of interest win = zeros(size(n)); L=13;%start with all zeros list of samples win(find(n>=0 & n<=(L-1)))=1; % replace zeros with ones to make pulse stem(n,win,'filled'); %make the plot OHIO S ATE T UNIVERSITY T . H . E ECE600 Part #1 20 Slide # 22 Important Signals: Exponential exponential sequence exponential sequence 1.2 1 0.8 0.6 amplitude 0.4 0.2 0 0.2 0.4 0.6 3 2 1 0 1 2 3 4 5 6 7 sample number, n >> >> >> >> a=-0.5;n = [-3:1:7]; x = a. n;%make exponential signal x(find(n<0))=0;%include action of u[n] stem(n,x,'filled'); axis([-3.1 7.1 -0.2 2.2]) OHIO S ATE T UNIVERSITY T . H . E ECE600 Part #1 Slide # 23 Important Signals: Sinusoid sinusoidal sequence sinusoidal sequence 10 8 6 4 amplitude 2 0 2 4 6 8 10 0 10 20 30 & 40 50 60 sample number, n >> >> >> >> n = [0:63];T = 0.001;% 101 samples; 1000 samples per second A = 10; omega = 2*pi*60*T; phi=pi/4;% ampl., freq., init. phase x = A*cos(omega*n + phi); stem(n,x,'filled');axis([0 63 -10 10]); OHIO S ATE T UNIVERSITY T . H . E ECE600 Part #1 ' Slide # 24 Important Signals: Damped Sinusoid damped sinusoidal sequence E.g., to model free induction decay in magnetic resonance (1952 Nobel prize). damped sinusoidal sequence 2 1.5 1 amplitude 0.5 0 0.5 1 1.5 10 5 0 5 10 15 20 25 >> A = 2; r = 0.88; omega = pi/5;phi = 0; >> x = A*r. n .* cos(omega*n + phi); >> x(find(n<0))=0; stem(n,x,'filled');% include u[n] and plot ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E sample number, n 30 & ' Slide # 25 Euler's Identity & Complex Exponentials Euler's identity will (believe it or not) make our arithmetic simpler. It follows that More generally, $ "# ! # & # % ! % & ' % ! ! "# % ! & ! ' ( $ ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 26 Complex Exponential: Example Im 1 0 7 Re 0 1 7 n ! ! real Above, imaginary and the sequence is shown for one peroid, ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 27 Complex-valued Signals: The Quadrature Receiver Also: complex exponentials are eigenfunctions of linear time-invariant systems. ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 28 Input-Output Systems A discrete-time system is an operation, or rule, that maps an input discrete-time signal to an output discrete-time signal. x[n] T y[n] Examples normalize the sum of squares to one Dow-Jones 30-day moving average a C code or Matlab subroutine ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 29 Signal Operations consider three operations 1. Addition: sequences add term-wise 2. Translation or Shift: delay by samples advance by samples 3. Scaling: multiply each term in the sequence ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 30 Signal Operations: Example Example: Consider again the model illustrating CELP speech coding: x[n] b0 y[n] -a1 -a2 -a3 ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 31 System Properties: Linearity Linearity A system is linear if it satis es the two properties of additivity and homogeneity. additivity homogeneity If is the response to , then is the response to for all scalars and all inputs Consequence! ECE600 Part #1 Slide # 32 UNIVERSITY . OHIO S ATE T T . H . E System Properties: Time-Invariance Time Invariance (shift invariance or translation invariance) If is the response to , then is the response to for every xed delay (or advance) and all inputs . LTI: linear, time-invariant ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 33 Example A linear system satis es the principle of superposition. Example 1: Linear? Let x[n] = a1 x1[n] + a2 x2[n]. Then y[n] = n{a1 x1[n] + a2 x2[n] } = a1{n x1[n]} = a1 y1[n] + a2 y2[n] So from the definition, the system is linear. + a2{n x2[n]} Time-invariant? No, the input/output relationship is a time-varying gain. Formally, we can construct a counter-example.Let x[n] = n u[n] (a ramp). Then, output = [0, 1, 4, 9, 16, 25, ...] for n >= 0. However, the output due to a delayed input x[n-2] is the sequence n { (n-2) u[n-2] }, so output = [0, 0, 0, 3, 8, 15, 24, ...]. Hence, shifting the input does not simply shift the output. ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 34 Example Example 2: Linear? max No. The system is neither additive nor homogeneous. One counterexample suffices to demonstrate that the system is not linear. Let x[n] = {0, 1, 2, 3, 4} for n=0,...,4 and x[n]=0 elsewhere. With input x[n], the output for n=0,...,8 is y[n] = {0, 1, 2, 3, 4, 4, 4, 0, 0}. However, for input -2x[n], the output is for n=0,...,8 is the sequence {0, 0, 0, -2, -4, 0, 0, 0, 0}, which is not equal to -2y[n]. Time-invariant? Yes, by direct application of the definition. Let x[n] produce output y[n]. Consider the input g[n] = x[n-d]. Then, g[n] produces output max{ g[n], g[n-1], g[n-2] } = max{ x[n-d], x[n-d], x[n-d] } which is the sequence y[n-d]. OHIO S ATE T UNIVERSITY T . H . E ECE600 Part #1 Slide # 35 System Properties: Impulse Response Impulse response The zero state output of a system resulting from input of a unit sample sequence is called the impulse response, . The impulse response completely characterizes the input-output relationship for a linear time-invariant (LTI) system. ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 36 Difference Equations Linear, constant coef cient difference equations are LTI systems Example 1: x[n] + A delay y[n] ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 37 Difference Equations Linear, constant coef cient difference equations are LTI systems Example 2: x[n] b0 b1 b2 b3 + + + + y[n] tapped delay line or linear transversal lter ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 38 Difference Equations: IIR Example Example 1: Recurse the IIR example. Let and ; therefore, Is this system stable? 0 1 2 3 4 5 1 0 0 0 0 0 0 1 . ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 39 Difference Equations: FIR Example Example 2: Recurse the FIR example. Let ; therefore, . 1 0 0 0 0 0 Is this system stable? 0 1 2 3 4 5 ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 40 Convolution Fundamental Property of LTI Systems: The zero state output of a LTI system can be completely described using the impulse response all Why is this? ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 41 System Properties: Stability Stability A system is bounded input, bounded output (BIBO) stable, if every bounded input produces a bounded output. For LTI systems, stability is equivalent to ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 42 System Properties: Stability Proof Suf ciency: Assume for all . Then, and let Necessity: Assume diverges. Then, for sgn , we have , but is unbounded. ECE600 Part #1 Slide # 43 OHIO S ATE T UNIVERSITY T . H . E System Properties: Causality Causality A system is causal if the output at any time index does not depend on the values of the input signal for indices . For LTI systems, causality is equivalent to for all For off-line data processing, non-causal lters can be useful. ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 44 Convolution: Properties We Use A LTI system performs convolution of the input with the system impulse response. This is the nature of LTI systems. A signal, , describes the system's input/output behavior. Convolution commutes ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 45 Convolution properties, continued Convolution associates Convolution distributes ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 46 Convolution properties, continued identity delay implies ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 47 Computing Convolution 1. Let the computer do it. In Matlab, >> y = conv(h,x); 2. In closed form. The Z-transform or state-space description can provide convenient solution techniques for difference equations. An explicit solution can be valuable in the analysis and design of systems. 3. Graphical convolution. Flip, shift, multiply, add. Very useful for a fast, qualitative, visual interpretation. 4. Fast computation using FFTs. Covered later in this course. ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 48 Example: ECE600 Part #1 ; zero for and . Computing Convolution, continued 5. Matrix-vector multiply (for nite duration signals). OHIO S ATE T UNIVERSITY T . H . E Slide # 49 Matlab Example >> >> >> >> >> >> >> >> n = [0:1:31]; x = cos(0.2*pi*n).';%.' turns row into column h = [-1 3 3 -1 ].'; ToepX = convmtx(x,length(h)); % (see 'help convmtx') y = ToepX*h; % construct a plot stem([0:length(y)-1], y,'filled'); Convolution Example 6 4 Output amplitude 2 0 2 4 6 0 5 10 15 20 25 30 35 Sample index, n ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 50 Application: Correlation 5 80 60 40 20 0 0 20 40 60 5 50 0 50 80 100 50 0 50 100 5 80 60 40 20 0 0 20 40 60 5 50 0 50 80 100 50 0 50 100 >> Rxv = xcorr(x,v) ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 51 ECE600 Part #1 . . . . . . . . . Example: Assume ; is known for Solve for the unknown impulse response, , . . . . ; . . . Application: FIR System Identi cation is measured. OHIO S ATE T UNIVERSITY T . H . E Slide # 52 Least Squares Solution >> h = X y; % or ... >> h = pinv(X)*y; % or ... >> h = inv(X'*X)*X'*y; ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 53 . . ECE600 Part #1 . . Equation Error Method for IIR . . . . . . . . . . OHIO S ATE T UNIVERSITY T . H . E Slide # 54 State Space Representations An LTI system may be written as a set of coupled rst-order difference equations that describe how the system evolves in time. The state of the system, , is the minimal set of signals representing the memory of the past necessary to determine the future. ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 55 State Space: Total Response A system's output may be decomposed as the response due to the initial stored energy plus the response due to the input alone. zero-input resp. zero-state resp. where ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 56 State Space Example: FIR Consider a third-order non-recursive system: x[n] b0 b1 b2 b3 + + + + y[n] Let OHIO S ATE T UNIVERSITY T . H . E ECE600 Part #1 Slide # 57 State Space Example: IIR For the direct form II realization, x[n] b0 y[n] a1 b1 a2 b2 OHIO S ATE T UNIVERSITY T . H . E ECE600 Part #1 Slide # 58 Linear Systems Review: Frequency Domain In time domain: convolution For LTI systems, we decomposed the input as a weighted sum of delayed impulses. We learned that the (zero state) output is weighted sum of delayed impulse responses. Frequency response: Following Fourier, we now shall represent the input as a sum of sines and cosines. We shall learn that for LTI systems the output is the superposition of responses due to each sinusoidal component. ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 59 Eigenfunctions of LTI Systems An analogy: x 1 Ax x x 2 LTI System ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 60 Let ECE600 Part #1 & Frequency Response % magnitude response & a complex scalar phase response OHIO S ATE T UNIVERSITY T . H . E Slide # 61 Sinusoidal Steady State The concepts of linearity, time-invariance and zero-state output gave us convolution, which is the central idea in the time-domain study of the input/output behavior of linear systems. Now consider the response to an everlasting sinusoidal signal: ! "# $ Real $ Real $ "# "# $ % % $ & ! "# $ & % Real % & ! ! ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 62 IIR Example Example 1: Consider a recursive LTI system: delay x[n] + 0.8 radians y[n] OHIO S ATE T UNIVERSITY T . H . E ECE600 Part #1 Slide # 63 IIR Example, continued IIR Example: y[n]=cos(pi n/20) + 0.8y[n 1] 5 transient 4 steady state Magnitude change 3 y[n] 2 amplitude 1 x[n] 0 1 2 Phase change 3 4 5 0 10 20 30 40 50 60 70 80 90 sample number, n ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 64 IIR Example, continued 15 10 Magnitude (dB) 5 0 5 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Normalized Angular Frequency ( rads/sample) 0.8 0.9 1 0 10 Phase (degrees) 20 30 40 50 60 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Normalized Angular Frequency ( rads/sample) 0.8 0.9 1 >> help freqz >> freqz([1],[1 -0.8]) ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 65 FIR Example % This is a second order FIR lter, and hence is a linear time-invariant system. We have that To nd , we start with the de nition % % % % % & Thus, and . In M ATLAB >> b = [0.5 1 0.5];a=1; >> freqz(b,a); ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E % Slide # 66 FIR Example, continued 2 1.5 1 0.5 0 Magnitude response 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 normalized frequency 0 -1 -2 -3 -4 0 Phase response 0.1 0.2 0.3 0.4 0.5 0.6 normalized frequency 0.7 0.8 0.9 1 For we have , and . Therefore, & & ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 67 Who Was Fourier? The Fourier transform dates to J.-B.J. Fourier , who at age 39 presented a manuscript, Theory of the Propagation of Heat in Solid Bodies, at the Institut de France on December 21, 1807. Fourier had reduced his problem to expressing an even function as a possibly in nite sum of cosines. Further, he gave a simple means of nding the series coef cients. Fourier's claim was controversial, to say the least. Years earlier, Daniel Bernoulli (1700-1782) had proposed in nite sums of trigonometric functions for modeling the vibrating string. Bernoulli's claims had been summarily dismissed by Euler (1707-1783). Likewise, the committee that reviewed Fourier's manuscript (Laplace, Lagrange, Lacroix, and Monge) gave an unenthusiastic report, written by Poisson. Lagrange (1736-1813) was later to make his objections explicit. Well into the 1820s, Fourier series would remain suspect. In January 1829, at the age of 23, Gustav Dirichlet gave the rst correct proof for the validity of Fourier series. Cauchy in the 1820s, Riemann in the 1860s, and Lebesgue in the 1900s were each to expand and clarify the meaning of integration. Fourier's manuscript precipitated a crisis in mathematics; the nineteenth century witnessed a reconstruction of the foundations of calculus, giving rise to the new discipline of analysis. Jean-Baptiste Joseph Fourier (1768 1830) was the son of a French tailor. Educated in a monastery, he left to engage in mathematical and revolutionary activities. He accompanied Napoleon to Egypt in 1798 and was later appointed prefect of the Department of Isere in southern France. His work in the mathematical theory of heat was a landmark in mathematical physics. Peter Gustav Lejeune Dirichlet (1805-1859) was born in the Rhineland and taught at Berlin for almost thirty years before going to Gottingen as Gauss' successor. He made fundamental contributions to number theory and analysis. Among his students was Leopold Kronecker (1823-1891). After making a fortune before he was thirty, he returned to mathematics. Kronecker is known for his work in algebra and number theory. ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 68 Frequency Response We have learned that the steady state output of a LTI system due to an exponential input is the same sequence, only scaled by the (complex) scalar . The scalar is given by ! and is called the frequency response of the LTI system. The frequency response describes the change in phase and amplitude of a complex exponential sequence as a function of the radian frequency, . For a single frequency, the frequency response is a complex number. Expressing this number as a magnitude and phase gives us the magnitude response and phase response % % & magnitude response phase response ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 69 Discrete Time Fourier Transform (DTFT) ! IDTFT or synthesis % ! DTFT or analysis ! % The notation The function denotes a DTFT pair. is the DTFT (spectrum) of the sequence % . magnitude spectrum phase spectrum & " The function any integer . is periodic in with period since $ for ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 70 Round Trip Ticket ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 71 Properties of the DTFT frequency response The impulse response, DTFT pair: and the frequency response are a & ! frequency response ! ! % % impulse response Linearity % ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 72 Properties of the DTFT, continued Symmetry For real valued, the discrete time Fourier transform is conjugate symmetric: % magnitude is an even function % Time shift frequency. A shift in time yields a linear phase factor changes the phase in % . Consequently, phase is an odd function & % ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 73 Example: Rational System Use only linearity and shift property to derive the frequency response for an arbitrary nite order, linear, constant coef cient difference equation: By linearity and shift, & % Grouping terms, % & % % Finally, % % ECE600 Part #1 >> freqz( OHIO [b0 b1 b2 b3], [1 a1 a2 a3] );%third order example S ATE T UNIVERSITY T . H . E Slide # 74 Properties of the DTFT, continued Modulation A shift in frequency modulates the time signal. ! % $ % Time reversal A reversal in time ( ipping) conjugates in frequency. Parseval's relation energy Energy is preserved by the transform. % % ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 75 Properties of the DTFT, continued Convolution Convolution in time is multiplication in frequency. ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 76 Properties of the DTFT, continued Windowing Multiplication of sequences corresponds to periodic convolution in the frequency domain. % Multiply L=7 1 periodic convolution 7 6 % 5 amplitude DTFTw[n] 0.8 4 amplitude w[n] 0.6 3 0.4 2 1 0.2 0 0 1 2 /L 2 0 2 0.2 2 0 2 4 6 8 2 sample number, n radians per sample ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 77 Rectangular Window ! " Let else Recall ! Dirichlet kernel is transform for ECE600 Part #1 on to . OHIO S ATE T UNIVERSITY T . H . E Slide # 78 Graphical View ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 79 Windowing Applications FIR design by windowed impulse response Sidelobe suppression in spectrum estimation Sidelobe suppression in Fourier imaging: CAT, radar, et al. Rayleigh resolution for optical apertures Beam pattern shaping by rolled surfaces Etc. ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 80 Sampling and Reconstruction Relating the continuous and discrete signal representations ... Outline: Ideal Sampling (ADC): CT DT CT Ideal Reconstruction (DAC): DT Implementation Issues in practice, same concept appears in many forms... application independent example sampling variable, increment ECG time sec sensor array spacing m radar frequency MHz image distance m Fourier transform variable, frequency angle time spatial frequency ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 81 Audio Example in Matlab fs = 1000; %1kHz sampling rate nT = [ 0 : 1/fs : 0.600 ];% 600 milliseconds for f = 100 : 100 : 1400; % 0.1kHz to 1.4kHz in 100Hz steps x = sin( 2* pi * f * nT + pi/4 ); sound(x,fs) pause(1.66) end Apparent frequency of sampled tone (Hz) 600 500 400 300 200 100 0 0 200 400 600 800 1000 1200 Frequency of sampled tone (Hz) 1400 ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 82 Sampling: Time Domain View A basic intuition for aliasing: 1 amplitude 0.5 0 0.5 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 time (seconds) 1 amplitude 0.5 0 0.5 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 time (seconds) ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 83 Aliasing: Simple Trigonometry ' & & ' ' ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 84 A Spinning Wheel Analogy time, t t=nT sample, n n=0 T = sampling interval t=nT n=1 t=nT n=2 t=nT n=3 Note: has units of radians per sample. OHIO S ATE T UNIVERSITY T . H . E ECE600 Part #1 Slide # 85 Resolving Ambiguity To resolve ambiguity, restrict Fundamental Interval: to some interval of Hz Nyquist rule: let sampling rate exceed twice the highest frequency of the signal (Shannon-Whittaker-Nyquist [1915] -Kotelnikov). ECE600 Part #1 max more than two samples per cycle OHIO S ATE T UNIVERSITY T . H . E Slide # 86 CT-FT and DT-FT How is the continuous-time Fourier transform of the signal related to the discrete-time Fourier transform of the samples? & % ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 87 Let ECE600 Part #1 ' & Derivation ' ' ' % % % & ' % ' % OHIO S ATE T UNIVERSITY T . H . E Slide # 88 & Graphical Interpretation The in nite sum of aliased images in frequency & % ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 89 Aliasing: the book-keeping 1. Counting frequency modulo A negative frequency? 100 Hz 1 0.5 0 0.5 1 1 0.5 0 0.5 1 0 5 10 15 20 0 5 10 15 20 100 Hz amplitude amplitude ECE600 Part #1 time, msec time, msec OHIO S ATE T UNIVERSITY T . H . E Slide # 90 Aliasing: the book-keeping 2. A graphical representation and Nyquist zones ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 91 Aliasing Tones: Example Let the sampling rate be Hz. The baseband version of the sampled signal is: ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 92 Aliasing: Example ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 93 Bandpass Sampling Sample at greater than twice the bandpass bandwidth. Generally, ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 94 Application: Digital Down Conversion Quarter-rate down conversion Generally, ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 95 Ideal Reconstruction ! 1 5 % 0.8 4 3 0.6 2 amplitude 0.4 amplitude 1 0 0.2 1 0 2 3 0.2 4 0.4 6 4 2 0 time (sec) 2 4 6 5 2.5 2 1.5 1 0.5 0 time (sec) 0.5 1 1.5 2 2.5 ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 96 Ideal Reconstruction, continued Ideal sinc interpolation lter ' ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 97 Recap ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 98 Discrete-Time Filtering x (t) c ideal sampler digital filter ideal interpolator y (t) c DT input spectrum DT LSI lter output reconstruction & ! ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 99 A/D & D/A: Practical Issues x (t) c anti alias filter sample and hold T A/D converter T x(n) DT system y(n) y (n) D/A converter T da recon struction filter G (j ) r y (t) c ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 100 Anti-Alias Filter ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 101 A/D Converters R (hold) (sample) C + - gnd V in + V ref R1 + + - MSB R 2 . . . _ V ref . . . . . . LSB digital logic B bits R 2N -2 + - fast clock analog input + comparator digital filter and downsample digital output ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 102 Quantization quantized output 3 2 clipping 3.5 1 2 3 4 2.5 input ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 103 Quantization Noise ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 104 Finite Precision: Over ow x 10 4 4 2 0 2 4 0 x 10 4 20 40 60 80 100 120 140 160 180 200 5 4 3 2 1 0 1 0 20 40 60 80 100 120 sample number 140 160 180 200 ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 105 D/A Converter V ref 2R R ... 2R R ... 2R 2R Rf + gnd Vout ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 106 Reconstruction Filter p(t) 1 t T T y 2 y (t) da y0 y 1 y y 3 4 t ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 107 Decimation 8 7 7 6 6 5 5 4 x(n) 4 y(n) 3 3 2 2 1 0 5 10 n 15 20 25 0 0 1 0 2 4 6 n 8 10 12 Linear? yes no OHIO S ATE T UNIVERSITY T . H . E Shift-invariant? ECE600 Part #1 Slide # 108 Decimation: frequency-domain % Example: ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 109 Decimation: frequency-domain ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 110 Interpolation 7 7 6 6 5 5 4 4 x(n) 3 y(n) 0 2 4 6 n 8 10 12 3 2 2 1 1 0 0 1 1 0 5 10 15 n 20 25 30 35 otherwise Linear? Shift-invariant? yes no OHIO S ATE T UNIVERSITY T . H . E ECE600 Part #1 Slide # 111 Interpolation: frequency-domain ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 112 Fractional Rate Example: 32 kHz to 48 kHz ECE600 Part #1 OHIO S ATE T UNIVERSITY T . H . E Slide # 113

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