Radar Systems Engineering Lecture 3 Review of Signals, Systems and Digital Signal Processing

Transcription

1 Radar Systems Engineering Lecture Review of Signals, Systems and Digital Signal Processing Dr. Robert M. O Donnell Guest Lecturer Radar Systems Course Review Signals, Systems & DSP //

2 Block Diagram of Radar System Target Radar Cross Section Propagation Medium Antenna T / R Switch Power Amplifier Transmitter Waveform Generation Signal Processor Computer Received Signal Receiver A / D Converter Pulse Compression Clutter Rejection (Doppler Filtering) Signal Strength General Purpose Computer Time Console / Displays Tracking Parameter Estimation Thresholding Detection Data Recording Radar Systems Course Review Signals, Systems & DSP // Application of Signals and Systems, and Digital Signal Processing Algorithms to the Received Radar Signals Result in Optimum Target Detection

3 Reasons for Review Lecture Signals and systems, and digital signal processing are usually one semester advanced undergraduate courses for electrical engineering majors In no way will this + hour lecture to justice to this large amount of material The lecture will present an overview of the material from these two courses that will be useful for understanding the overall Radar Systems Engineering course Goal of lecture- Give non EE majors a quick view of material; they may wish to study in more depth to enhance their understanding of this course. UC Berkeley has an excellent, free, video Signals and Systems course (ECE ) online at //webcast.berkeley.edu Given in Spring 7 Radar Systems Course Review Signals, Systems & DSP //

4 Signal Processing Signal processing is the manipulation, analysis and interpretation of signals. Signal processing includes: Adaptive filtering / thresholding Spectrum analysis Pulse compression Doppler filtering Image enhancement Adaptive antenna beam forming, and A lot of other non-radar stuff ( Image processing, speech processing, etc. It involves the collection, storage and transformation of data Analog and digital signal processing A lot of processing horsepower is usually required Radar Systems Course 4 Review Signals, Systems & DSP //

5 Outline Continuous Signals Sampled Data and Discrete Time Systems Discrete Fourier Transform (DFT) Fast Fourier Transform (FFT) Finite Impulse Response (FIR) Filters Weighting of Filters Radar Systems Course 5 Review Signals, Systems & DSP //

6 Continuous Time Signal x( t) t Examples: x x x ( t) = sin( πt) 79cos( πt) () t () 5 t = = t t t + 5t Radar Systems Course 6 Review Signals, Systems & DSP //

7 Continuous Time Signal x () t x( t) Non-periodic Periodic ( t + Δt) x( t) x = t t Types of continuous time signals Periodic or Non-periodic Radar Systems Course 7 Review Signals, Systems & DSP //

8 Continuous Time Signal Re [ x() t ] Im x [ ( t) ] t t x( t) is a complex periodic signal Types of continuous time signals Periodic or Non-periodic Real or Complex Radar signals are complex Radar Systems Course 8 Review Signals, Systems & DSP //

9 Continuous, Linear, Time Invariant Systems Continuous Linear Time Invariant System x () t y( t) y ( t) = T[ x( t) ] T = Operator Continuous If x() t and y( t) are continuous time functions, the system is a continuous time system Linear If the system satisfies [ α x ( t) + β x ( t) ] = α y ( t) + y ( t) T β Time Invariant If a time shift in the input causes the same time shift in the output Radar Systems Course 9 Review Signals, Systems & DSP //

10 Linear Time Invariant Systems (Delta Function) Continuous Linear Time Invariant System x () t y( t) Continuous Linear Time Invariant System δ ( t) h( t) Properties of Delta Function δ () t = δ t t = () t dt = The impulse response h t is the response of the system when the input is δ t () ( ) Radar Systems Course Review Signals, Systems & DSP //

11 Linear Time Invariant Systems Continuous Linear Time Invariant System x () t y( t) Continuous Linear Time Invariant System δ ( t) h( t) Definition : Convolution of Two Functions x () t x () t x () τ x ( t τ) dτ Reversed and Shifted Radar Systems Course Review Signals, Systems & DSP //

12 Linear Time Invariant Systems Continuous Linear Time Invariant System x () t y( t) Continuous Linear Time Invariant System δ ( t) h( t) y Convolution of and () t x() t h() t = x() τ h( t τ) dτ = x ( t) h( t) The output of any continuous time, linear, time-invariant (LTI) system is the convolution of the input x() t with the impulse response of the system h t ( ) Radar Systems Course Review Signals, Systems & DSP //

13 Why not Analog Sensors and Calculation Systems? Voltmeter Disadvantages Measurement Repeatability Environmental Sensitivity Size Complexity Courtesy of Hannes Grobe Cost Slide Rule Courtesy of oschene Courtesy of US Navy Torpedo Data Computer (94s) Radar Systems Course Review Signals, Systems & DSP //

14 Outline Continuous Signals and Systems Sampled Data and Discrete Time Systems General properties A/D Conversion Sampling Theorem and Aliasing Convolution of Discrete Time Signals Fourier Properties of Signals Continuous vs. Discrete Periodic vs. Aperiodic Discrete Fourier Transform (DFT) Fast Fourier Transform (FFT) Finite Impulse Response (FIR) Filters Weighting of Filters Radar Systems Course 4 Review Signals, Systems & DSP //

15 Sampled Data Systems Digital signal processing deals with sampled data Digital processing differs from processing continuous (analog) signals Digital Samples are obtained with a Sample and Hold (S/H) Amplifier followed by an Analog-to-Digital (A/D) converter Sampling rate Word length Radar Systems Course 5 Review Signals, Systems & DSP //

16 Waveform Sampling Sampling converts a continuous signal into a sequence of numbers Continuous-time System x ( t) Discrete-time System x[ n] A/D Converter Radar signals are complex Radar Systems Course 6 Review Signals, Systems & DSP //

17 Outline Continuous Signals and Systems Sampled Data and Discrete Time Systems General properties A/D Conversion Sampling Theorem and Aliasing Convolution of Discrete Time Signals Fourier Properties of Signals Continuous vs. Discrete Periodic vs. Aperiodic Discrete Fourier Transform (DFT) Fast Fourier Transform (FFT) Finite Impulse Response (FIR) Filters Weighting of Filters Radar Systems Course 7 Review Signals, Systems & DSP //

18 Ideal Analog to Digital (A/D) Converter V OUT V IN A/D Converter V OUT P(V ERROR q ) V FS q V FS V IN q σ VERROR = Radar Systems Course 8 Review Signals, Systems & DSP // q q q q V ERROR = V OUT V V IN IN

19 Non-Perfect Nature of A/D Converters Non- Monotonic Output Output Missing Bit Actual Input Offset Input Gain Missing bits Monotonicity Offset Nonlinearity Missing bits Ideal Radar Systems Course 9 Review Signals, Systems & DSP //

20 Single Tone A/D Converter Testing Fundamental Power Level (dbm) Spur Free Dynamic Range (SFDR) Highest Spur Frequency (MHz) For Ideal A/D S/N=6.N +.76 db Radar Systems Course Review Signals, Systems & DSP //

21 A/D Word Length A / D output is signed N bit integers Twos complement arithmetic Quantization noise power = Signal-to-noise ratio SNR, S / N must fit o within the word length: S = maximum signal power (target, jamming, clutter) = thermal noise power in A / D input N o α 4 Typically, to reduce clipping (limiting) Required word length: / ( ), L > αs / < N ( SNR / 6). L > DB + o A/D Saturation Head Room (~ db) Maximum Signal Receiver Dynamic Range Noise Signal Foot Room (~ db) Noise Quantization SNRDB = log SNR Radar Systems Course Review Signals, Systems & DSP //

22 Outline Continuous Signals and Systems Sampled Data and Discrete Time Systems General properties A/D Conversion Sampling Theorem and Aliasing Convolution of Discrete Time Signals Fourier Properties of Signals Continuous vs. Discrete Periodic vs. Aperiodic Discrete Fourier Transform (DFT) Fast Fourier Transform (FFT) Finite Impulse Response (FIR) Filters Weighting of Filters Radar Systems Course Review Signals, Systems & DSP //

23 Waveform Sampling Sampling converts a continuous signal into a sequence of numbers Continuous-time System x ( t) Discrete-time System x[ n] A/D Converter Radar signals are complex Radar Systems Course Review Signals, Systems & DSP //

24 Sampling - Overview Sampling Theorem constraint (a.k.a. Nyquist criterion) to prevent aliasing : For continuous aperiodic signals: F s B F s = Sampling Frequency Nyquist criterion: Permits reconstruction via a low pass filtering Eliminates Aliasing Radar Systems Course 4 Review Signals, Systems & DSP //

25 Signal Sampling Issues Signal Reconstruction F s > B LPF X( F) B X c ( F) F s F s Elimination of Aliasing X( F) Overlapping, Aliased Spectra F s < B F s Fs F s Fs 4Fs Radar Systems Course 5 Review Signals, Systems & DSP //

26 The Sampling Theorem If (t) is strictly band limited, x c X (F) = for F > B then, x c (t) may be uniquely recovered from its samples B F S = π T S B The frequency is called the Nyquist frequency, and the minimum sampling frequency, F S = B, is called the Nyquist rate x[ n] if Radar Systems Course 6 Review Signals, Systems & DSP //

27 Spectrum of a Sampled Signal Sampling periodically replicates the spectrum Fourier transform of a sampled signal is periodic X c ( ) ( ) x c ( ) [ n] If F and X F are the spectra of t and x X c ( F) = x ( t) c e jπft dt ( ) = g( t) δ( t nt) X F n= e jπft dt X c ( F) = n= x jπ Fs [ n] e nf / X( F) F s Fs F s Fs Radar Systems Course 7 Review Signals, Systems & DSP //

28 Distortion of a Signal Spectrum by Aliasing Assume band limited so that x c (t) X (f ) =, for f > B X c ( F) x c (t) F S B If is sampled with F S If is sampled with x c (t) F S < B B B X( F) / T S X( F) / T S B B No Aliasing FS Aliased parts of spectrum F F F S F S / F S / F S F Radar Systems Course 8 Review Signals, Systems & DSP //

29 Effect of Sampling Rate on Frequency..5 x c (t) -5 5 t (sec) Sampled Signal Its Fourier Transform x x c Continuous Signal (t) A t = e, A > A T n AT [ n] = x (nt) = e = ( e ) X Its Fourier Transform X c (F) = A A + (πf) n n AT c = a a = e, FS = ( ω) = x[ n] n= e j ω n a = acosω + a, ω = π T F F S X = T l= ( F) X ( F lf ) = Xˆ ( F) = c c xˆ c(t) Reconstructed Signal c Inverse Fourier Transform T FS X( F) F FS F > Radar Systems Course 9 Review Signals, Systems & DSP // Adapted from Proakis and Manolakis, Reference

30 Spectrum of Reconstructed Signal Signal Frequency Spectrum Continuous Signal..5 x c (t) X c ( F) Sampled Signal x [ n] = x ( nt) c t (sec) T = sec Frequency (Hz) F S = Hz Sampled Signal x Radar Systems Course Review Signals, Systems & DSP // [ n] = x ( nt) c t (sec) t (sec) T = sec Frequency (Hz) F S = Hz Frequency (Hz) Adapted from Proakis and Manolakis, Reference 4

31 Outline Continuous Signals and Systems Sampled Data and Discrete Time Systems General properties A/D Conversion Sampling Theorem and Aliasing Convolution of Discrete Time Signals Fourier Properties of Signals Continuous vs. Discrete Periodic vs. Aperiodic Discrete Fourier Transform (DFT) Fast Fourier Transform (FFT) Finite Impulse Response (FIR) Filters Weighting of Filters Radar Systems Course Review Signals, Systems & DSP //

32 Convolution for Discrete Time Systems Continuous-time System x () t y = Continuous Linear Time Invariant System x ( t) y( t) () t h() τ x( t τ) dτ Discrete-time System x[ n] x[ n] y Discrete Linear Time Invariant System y[ n] [ n] h[ k] x[ n k] = k= Radar Systems Course Review Signals, Systems & DSP //

33 y Graphical Implementation of Convolution [ n] h[ k] x[ n k] = x[ k] h[ n k] = k= k= Example: h[ k]= x[ k] = x [ ] h[ k] Step : Plot the sequences, k and Radar Systems Course Review Signals, Systems & DSP //

34 y Graphical Implementation of Convolution [ n] h[ k] x[ n k] = x[ k] h[ n k] = k= k= Example: h[ k]= x[ k] = h[ k] = - - Step : Take one of the sequences and time reverse it Radar Systems Course 4 Review Signals, Systems & DSP //

35 y Graphical Implementation of Convolution [ n] h[ k] x[ n k] = x[ k] h[ n k] = k= k= Example: h[ k]= x[ k] = h[ k] = - - [ ] Step : Shift h k by n, yielding a shift to the left a shift to the right n < h[ n k] = n > n-,n-,n Radar Systems Course 5 Review Signals, Systems & DSP //

36 y Graphical Implementation of Convolution [ n] h[ k] x[ n k] = x[ k] h[ n k] = k= k= Example: h[ k]= x[ k] = h[ k] = - - Step 4 : For each value of n,multiply the sequences x k and h[ n k] ; and add products together for all values of to produce y n [ ] [ ] k Radar Systems Course 6 Review Signals, Systems & DSP //

37 Example: h[ k]= y Graphical Implementation of Convolution [ n] h[ k] x[ n k] = x[ k] h[ n k] = k= x[ k] = k= for n = h[ k] = - - h[ k]= No overlap = Radar Systems Course 7 Review Signals, Systems & DSP // x[ k]= y [ n] y[ n] Output of Convolution Sample Number

38 Example: h[ k]= y Graphical Implementation of Convolution [ n] h[ k] x[ n k] = x[ k] h[ n k] = k= x[ k] = k= for n = h[ k] = - h[ k]= One sample overlaps = (x) = Radar Systems Course 8 Review Signals, Systems & DSP // x[ k]= y [ n] y[ n] Output of Convolution Sample Number

39 Example: h[ k]= y Graphical Implementation of Convolution [ n] h[ k] x[ n k] = x[ k] h[ n k] = k= x[ k] = 4 k= 4 5 for n = h[ k] = h[ k]= Two samples overlaps = (x)+(x) = 4 Radar Systems Course 9 Review Signals, Systems & DSP // x[ k]= y [ n] y[ n] Output of Convolution Sample Number

40 Example: h[ k]= y Graphical Implementation of Convolution [ n] h[ k] x[ n k] = x[ k] h[ n k] = k= x[ k] = k= for n = h[ k] = h[ k]= Three samples overlaps = (x)+(x)+(4x) = Radar Systems Course 4 Review Signals, Systems & DSP // x[ k]= y [ n] y[ n] Output of Convolution Sample Number

41 Example: h[ k]= y Graphical Implementation of Convolution [ n] h[ k] x[ n k] = x[ k] h[ n k] = k= x[ k] = 4 k= 4 5 for n = 4 h[ 4 k] = 4 h[ 4 k]= Three samples overlaps = (x)+(4x)+(x) = 7 Radar Systems Course 4 Review Signals, Systems & DSP // x[ k]= y [ n] y[ n] Output of Convolution Sample Number

42 Example: h[ k]= y Graphical Implementation of Convolution [ n] h[ k] x[ n k] = x[ k] h[ n k] = k= x[ k] = 4 k= 4 5 for n = 5 h[ 5 k] = 4 5 Three samples overlaps = (4x)+(x)+(x) = 7 Radar Systems Course 4 Review Signals, Systems & DSP // x[ k]= h[ 5 k]= y [ n] y[ n] Output of Convolution Sample Number

43 Example: h[ k]= y Graphical Implementation of Convolution [ n] h[ k] x[ n k] = x[ k] h[ n k] = k= x[ k] = k= for n = 6 h[ 6 k] = h[ 6 k]= Two samples overlaps = (x)+(x) = Radar Systems Course 4 Review Signals, Systems & DSP // x[ k]= y [ n] y[ n] Output of Convolution Sample Number

44 Example: h[ k]= y Graphical Implementation of Convolution [ n] h[ k] x[ n k] = x[ k] h[ n k] = k= x[ k] = 4 k= 4 5 for n = 7 h[ 7 k] = h[ 7 k]= One sample overlaps = (x) = Radar Systems Course 44 Review Signals, Systems & DSP // x[ k]= y [ n] y[ n] Output of Convolution Sample Number

45 Example: h[ k]= y Graphical Implementation of Convolution [ n] h[ k] x[ n k] = x[ k] h[ n k] = k= x[ k] = 4 k= 4 5 for n = 8 h[ 8 k] = h[ 8 k]= y No overlap = Radar Systems Course 45 Review Signals, Systems & DSP // x[ k]= [ n] y[ n] Output of Convolution Sample Number

46 Summary- Linear Discrete Time Systems Any Linear and Time-Invariant (LTI) system can be completely described by its impulse response sequence δ[ n] H h[ n] The output of any LTI can be determined using the convolution summation y [ n] h[ k] x[ n k], < n < = = k The impulse response provides the basis for the analysis of an LTI system in the time-domain The frequency response function provides the basis for the analysis of an LTI system in the frequency-domain Radar Systems Course 46 Review Signals, Systems & DSP // Adapted from MIT LL Lecture Series by D. Manolakis

47 Outline Continuous Signals and Systems Sampled Data and Discrete Time Systems General properties A/D Conversion Sampling Theorem and Aliasing Convolution of Discrete Time Signals Fourier Properties of Signals Continuous vs. Discrete Periodic vs. Aperiodic Discrete Fourier Transform (DFT) Fast Fourier Transform (FFT) Finite Impulse Response (FIR) Filters Weighting of Filters Radar Systems Course 47 Review Signals, Systems & DSP //

48 Frequency Analysis of Signals Decomposition of signals into their frequency components A series of sinusoids of complex exponentials The general nature of signals Continuous or discrete Aperiodic or periodic Radar echoes, from each transmitted pulse, are continuous and aperiodic, and are usually transformed into discrete signals by an A/D converter before further processing Complex signals Radar Systems Course 48 Review Signals, Systems & DSP //

49 Time and Frequency Domains Analysis Fourier Transform Time History Frequency Spectrum Time Domain Frequency Domain Inverse Fourier Transform Synthesis Radar Systems Course 49 Review Signals, Systems & DSP //

50 Fourier Properties of Signals Continuous-Time Signals Periodic Signals: Fourier Series Aperiodic Signals: Fourier Transform Discrete-Time Signals Periodic Signals: Fourier Series Aperiodic Signals: Fourier Transform Radar Systems Course 5 Review Signals, Systems & DSP //

51 Fourier Transform for Continuous-Time Aperiodic Signals Time Domain Continuous and Aperiodic Signals Frequency Domain Continuous and Aperiodic Signals x (t) X(F) X(F) t π = x(t) e j F t dt x(t) j π F t = X(F) e df F Adapted from Manolakis et al, Reference Radar Systems Course 5 Review Signals, Systems & DSP //

52 Fourier Properties of Signals Continuous-Time Signals Periodic Signals: Fourier Series Aperiodic Signals: Fourier Transform Discrete-Time Signals Periodic Signals: Fourier Series Aperiodic Signals: Fourier Transform Radar Systems Course 5 Review Signals, Systems & DSP //

53 Fourier Transform for Discrete-Time Aperiodic Signals Time Domain Discrete and Aperiodic Signals x[ n] Frequency Domain Continuous and Periodic Signals X[ ω] n 4 4 π π π π ω X( ω) = n= j ω n [ ] e X n x π [ n] = π X( ω)e j ω n dω Adapted from Malolakis et al, Reference Radar Systems Course 5 Review Signals, Systems & DSP //

54 Summary of Time to Frequency Domain Properties Continuous- Time Signals Discrete- Time Signals Time-Domain x() t Frequency-Domain c k Time-Domain x[] n Frequency-Domain c k Periodic Signals Fourier Series T P = T jπkft ck x() t e dt TP P TP Continuous and Periodic x() t t xt () = ce k k= F = TP jπkf t Discrete and Aperiodic XF ( ) F N c k N N = xne [] N n= π j kn N n N xn [] N = k= ce k π j kn N Discrete and Periodic Discrete and Periodic x[] n X( ω) N k Aperiodic Signals Fourier Transforms t 4 F 4 n π π π π ω jωn X( ω= ) xne [ ] jπft X( F) = x( t) e dt x() t X( F) e j π = Ft df Continuous and Aperiodic Continous and Aperiodic Discrete and Aperiodic Continous and Periodic n= jωn x[] n = X() ωe dω π π Adapted from Proakis and Manolakis, Reference Radar Systems Course 54 Review Signals, Systems & DSP //

55 Outline Continuous Signals and Systems Sampled Data and Discrete Time Systems Discrete Fourier Transform (DFT) Calculation Fast Fourier Transform (FFT) Finite Impulse Response (FIR) Filters Weighting of Filters Radar Systems Course 55 Review Signals, Systems & DSP //

56 N [ ] = x [ n] X k X X R I n= N Direct DFT Computation [ k] = x [ n] cos k n + x [ n] n= N [ k] = x [ n] sin k n x [ n] n= R W R kn N π N π N k N I I sin cos Aka Twiddle Factor W kn N = e π k n N π k n N π j k n / N. N evaluations of trigonometric functions. 4N real ( N complex) multiplications. 4N(N ) real ( N(N ) complex) additions 4. A number of indexing and addressing operations N Complex MADS MADS Multiply And Divides Adapted from MIT LL Lecture Series by D. Manolakis Radar Systems Course 56 Review Signals, Systems & DSP //

57 Outline Continuous Signals and Systems Sampled Data and Discrete Time Systems Discrete Fourier Transform (DFT) Fast Fourier Transform (FFT) Finite Impulse Response (FIR) Filters Weighting of Filters Radar Systems Course 57 Review Signals, Systems & DSP //

58 Fast Fourier Transform (FFT) An algorithm for each efficiently computing the Discrete Fourier Transform (DFT) and its inverse DFT O ( N ) MADS (Multiplies and Divides) FFT N O log N MADS FFT algorithm Development - Cooley / Tukey (965) Gauss (85) Many variations and efficiencies of the FFT algorithm exist Decimation in Time (input - bit reversed, output - natural order) Decimation in Frequency (input - natural order, output - bit reversed) The FFT calculation is broken down into a number of sequential stages, each stage consisting of a number of relatively small calculations called Butterflies Radar Systems Course 58 Review Signals, Systems & DSP //

59 Radix Decimation in Time FFT Algorithm N N π j N kn kn j k n / N [ ] [ ] k n / π x n e = x [ n] W k N W = e X k = n= n= N N Divide DFT of size N into two interleaved DFTs, each of size N/ Example will be N = = 8 Input to each DFT are even and odd x[ n] s, respectively Solve each stage recursively, until the size of the stage s DFT is. N nk nk [ ] = x[ n] W = x[ n] W + x[ n] X k = n= N l= g Radar Systems Course 59 Review Signals, Systems & DSP // N n N Even N n Odd N N lk (l+ )k lk k lk [] l W + h[] l W = g[] l W + W h[] l W N N N / N N / l= l= l= Even index and odd index terms of x [ n] N/ point DFT of h [ l] = H[ k] W nk N N/ point DFT of g [ l] = G[ k]

60 Radix Decimation in Time FFT Algorithm (continued) nk [ ] = G[ k] + W H[ k] X k Using the periodicity of the complex exponentials: N N G[ k] = G k + H[ k] = H k + And the following properties of the twiddle factors : W k+ (N / ) N then W = W k N k+ (N / ) N W H N / N A block diagram of this computational flow is graphically illustrated in the next chart for an 8 point FFT N = W k N k ( k + ( N / ) ) = W H(k) N Radar Systems Course 6 Review Signals, Systems & DSP //

61 8 Point Decimation in Time FFT Algorithm (After First Decimation) x[] x[] x[] 4 x[] 6 x[] x[] x[] 5 x[] 7 4 Point DFT 4 Point DFT G[ ] G[ ] G[ ] G[ ] H[ ] H[ ] H[ ] H[ ] X[ ] W 8 X [ ] W 8 X[ ] W 8 W 8 4 W 8 5 W 8 6 W 8 7 W 8 [ ] X [ ] X 4 [ ] [ ] X 5 X 6 [ ] X 7 Radar Systems Course 6 Review Signals, Systems & DSP //

62 Decimation of 4 Point into two point DFTs If N/ is even, n and h n may again be decimated G N This leads to: N N nk nk nk [ k] = g[ n] W = g[ n] W + g[ n] W G n= g [ ] [ ] N N / n Even N / N n Odd N / 4 nk k nk [ k] = g[ n] W + W g [ n + ] W n= N / 4 N / n= N / 4 x[] x[] x[] 4 x[] 6 Point DFT Point DFT W 4 W 4 W 4 W 4 [ ] G [ ] G [ ] G [ ] G Radar Systems Course 6 Review Signals, Systems & DSP //

63 Butterfly for Point DFT q[ k] q[] [ ] = q[ ] q[ ] Q + q [] Q[ ] = q[ ] q[ ] Q[ k] Now, Putting it all together.. Radar Systems Course 6 Review Signals, Systems & DSP //

64 x[] x[] 4 x[] x[] 6 x[] x[] 5 x[] x[] 7 Flow of 8-Point FFT (Radix - Decimation in Time Algorithm) W 8 W 8 W 8 W 8 W 8 W 8 W 8 W 8 W 8 W 8 W 8 W 8 X[ ] X[ ] X[ ] X[ ] X[ 4] X[ 5] X[ 6] X[ 7] Radar Systems Course 64 Review Signals, Systems & DSP //

65 Basic FFT Computation Flow Graph a Butterfly A = a + W r N b k n π W N = e r = k n j k n/ N Check over Each Butterfly takes MADS (Multiplies and Adds) Twiddle Factors (For 8 point FFT) π j / 8 π j / 4 W = e = W = e = e = j / W 8 8 b = e π r W N j / = j W 8 ( j) / ( ) Butterflies implies MADS vs. 64 MADS for 8 point DFT 5 point FFT more than times faster than 5 DFT 8 = B e = a W π j / 4 = r N Twiddle Factor b Radar Systems Course 65 Review Signals, Systems & DSP //

66 Computational Speed DFT vs. FFT Number of Complex Multiplications DFT FFT Number of points in Radix FFT 4 Lines Drawn Through Data Points Discrete Fourier Transform (O ~ N ) Fast Fourier Transform (O ~ N log N) Radar Systems Course 66 Review Signals, Systems & DSP // Adapted from Lyons, Reference

67 Fast Fourier Transform (FFT) - Summary Fast Fourier Transform (FFT) algorithms make possible the computation of DFT with O ((N/) log N) MADS as opposed to O N MADS Many other implementations of the FFT exist: Radix decimation in frequency algorithm Radar-Brenner algorithm Bluestein s algorithm Prime Factor algorithm The details of FFT algorithms are important to the designers of real-time DSP systems in software or hardware An interesting history of FFT algorithms Heideman, Johnson, and Burrus, Gauss and the History of FFT, IEEE ASSP Magazine, Vol., No. 4, pp. 4-, October 984 Radar Systems Course 67 Review Signals, Systems & DSP //

68 Outline Continuous Signals and Systems Sampled Data and Discrete Time Systems Discrete Fourier Transform (DFT) Fast Fourier Transform (FFT) Finite Impulse Response (FIR) Filters Weighting of Filters Radar Systems Course 68 Review Signals, Systems & DSP //

69 Finite and Infinite Response Filters Infinite Impulse Response (IIR) Filters Output of filter depends on past time history Example : M y[ n] = x[ n] + y[ n ] M M ( ) Finite Impulse Response (FIR) Filters Output depends on the finite past Example: DFT N π j k n / N [ ] x [ n] e X k = n= Radar Systems Course 69 Review Signals, Systems & DSP // Other examples: or y y N [ k] a[ k,n] x [ n] x[ ] = n= [ n] = x[ n,] x[ n,]

70 Four Basic Filter Types- An Idealization Ideal Low Pass Filter ( jw H e ) Ideal High Pass Filter ( jw H e ) π ω c ωc π ω π ωc ωc π ω Ideal Bandpass Filter ( jw H e ) Ideal Bandstop Filter ( jw H e ) π ω ω ω ω π ω π ω ω ω ω π ω Radar Systems Course 7 Review Signals, Systems & DSP //

71 Outline Continuous Signals and Systems Sampled Data and Discrete Time Systems Discrete Fourier Transform (DFT) Fast Fourier Transform (FFT) Finite Impulse Response (FIR) Filters Weighting of Filters Radar Systems Course 7 Review Signals, Systems & DSP //

72 Windowing / Weighting of Filters If we take a square pulse, sample it M times, and calculate the Fourier transform of this uniform rectangular window : W W M j ω M ( ) ( ) ( ω ) ω = j ω n e j M / sin M / e = = e j ω n= e sin( ω/ ) sin( ωm / ) ( ω) = π ω π sin( ω/ ) This is recognized as the sinc function which has db sidelobes If lower sidelobes are needed, at the cost of a widened pass band, one can multiply the elements of the pulse sequence with one of a number of weighting functions, which will adjust the sidelobes appropriately Radar Systems Course 7 Review Signals, Systems & DSP //

73 Commonly Used Window Functions Rectangular w[ n] =, n M, otherwise Bartlett (triangular) Hanning Hamming Blackman w[ n]= [ ] = w n w[ n] = w[ n] = n / M, n / M, otherwise otherwise n M / M / < n M ( π n / M).5.5 cos, n M, otherwise ( π n / M).54.46cos, n M, ( π n / M) +.8 cos ( 4 π n / ).4.5cos M, n, otherwise M Radar Systems Course 7 Review Signals, Systems & DSP //

74 Comparison of Common Windows Type of Window Rectangular Bartlett (triangular) Hanning Hamming Blackman Peak Sidelobe Amplitude (db) Approximate Width of Main Lobe ( M ) 4 π / + 8π / M 8π / M 8π / M π / M Radar Systems Course 74 Review Signals, Systems & DSP //

75 Comparison of Rectangular & Hamming Windows Rectangular Hamming log W(ω Normalized frequency (f = F/F s ) Radar Systems Course 75 Review Signals, Systems & DSP //

76 Summary A brief review of the prerequisite Signal & Systems, and Digital Signal Processing knowledge base for this radar course has been presented Viewers requiring a more in depth exposition of this material should consult the references at the end of the lecture The topics discussed were: Continuous signals and systems Sampled data and discrete time systems Discrete Fourier Transform (DFT) Fast Fourier Transform (FFT) Finite Impulse Response (FIR) filters Weighting of filters Radar Systems Course 76 Review Signals, Systems & DSP //

77 References. Proakis, J, G. and Manolakis, D. G., Digital Signal Processing, Principles, Algorithms, and Applications, Prentice Hall, Upper Saddle River, NJ, 4 th Ed., 7. Lyons, R. G., Understanding Digital Signal Processing, Prentice Hall, Upper Saddle River, NJ, nd Ed., 4. Hsu, H. P., Signals and Systems, McGraw Hill, New York, Hayes, M. H., Digital Signal Processing, McGraw Hill, New York, Oppenheim, A. V. et al, Discrete Time Signal Processing, Prentice Hall, Upper Saddle River, NJ, nd Ed., Boulet, B., Fundamentals of Signals and Systems, Prentice Hall, Upper Saddle River, NJ, nd Ed., 7. Richards, M. A., Fundamentals of Radar Signal Processing, McGraw Hill, New York, 5 8. Skolnik, M., Radar Handbook, McGraw Hill, New York, nd Ed., Skolnik, M., Introduction to Radar Systems, McGraw Hill, New York, rd Ed., Radar Systems Course 77 Review Signals, Systems & DSP //

78 Acknowledgements Dr Dimitris Manolakis Dr. Stephen C. Pohlig Dr William S. Song Radar Systems Course 78 Review Signals, Systems & DSP //

79 Homework Problems From Proakis and Manolakis, Reference Problems.,.7, 4.9a and b, 4. a and b, 6., 6.9 a and b, 8. and 8.8 Or And from Hays, Reference 4 Problems.4,.49,.54,.59,.46,.57,.58,.7,.8,.4, 6.44, 6.45 Radar Systems Course 79 Review Signals, Systems & DSP //