Ce diaporama a bien été signalé.
Nous utilisons votre profil LinkedIn et vos données d’activité pour vous proposer des publicités personnalisées et pertinentes. Vous pouvez changer vos préférences de publicités à tout moment.

Radar 2009 a 3 review of signals systems and dsp

1 244 vues

Publié le

Radar Systems Engineering - Lect 3

Publié dans : Ingénierie
  • Soyez le premier à commenter

Radar 2009 a 3 review of signals systems and dsp

  1. 1. IEEE New Hampshire Section Radar Systems Course 1 Review Signals, Systems & DSP 1/1/2010 IEEE AES Society Radar Systems Engineering Lecture 3 Review of Signals, Systems and Digital Signal Processing Dr. Robert M. O’Donnell IEEE New Hampshire Section Guest Lecturer
  2. 2. Radar Systems Course 2 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Block Diagram of Radar System Transmitter Waveform Generation Power Amplifier T / R Switch Propagation Medium Target Radar Cross Section Pulse Compression Receiver Clutter Rejection (Doppler Filtering) A / D Converter General Purpose Computer Tracking Data Recording Parameter Estimation Detection Signal Processor Computer ThresholdingConsole / Displays Antenna Received Signal Time Signal Strength Application of Signals and Systems, and Digital Signal Processing Algorithms to the Received Radar Signals Result in Optimum Target Detection
  3. 3. Radar Systems Course 3 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society 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 1+ 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 120) online at //webcast.berkeley.edu – http://webcast.berkeley.edu/course_details.php?seriesid=1906978405 – Given in Spring 2007
  4. 4. Radar Systems Course 4 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society 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
  5. 5. Radar Systems Course 5 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society 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
  6. 6. Radar Systems Course 6 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Continuous Time Signal ( )tx t0 ( ) ( ) ( ) ( ) ( ) 532 t25tttx 300t12tx t3cos79tsin100tx − +−= −= π−π= Examples:
  7. 7. Radar Systems Course 7 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Continuous Time Signal t0 ( )tx ( )tx t0 • Types of continuous time signals – Periodic or Non-periodic Non-periodic • • •• • • ( ) ( )txttx =Δ+ Periodic
  8. 8. Radar Systems Course 8 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Continuous Time Signal t • Types of continuous time signals – Periodic or Non-periodic – Real or Complex Radar signals are complex ( )[ ]txRe t0 0 ( )[ ]txIm • • • • • • • • • • • • is a complex periodic signal( )tx
  9. 9. Radar Systems Course 9 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Continuous, Linear, Time Invariant Systems Continuous Linear Time Invariant System ( )tx ( )ty • Continuous – If and are continuous time functions, the system is a continuous time system • Linear – If the system satisfies • Time Invariant – If a time shift in the input causes the same time shift in the output ( )tx ( ) ( )[ ] == TtxTty ( )ty ( ) ( )[ ] ( ) ( )tytytxtxT 2121 β+α=β+α Operator
  10. 10. Radar Systems Course 10 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Linear Time Invariant Systems (Delta Function) • The impulse response is the response of the system when the input is Continuous Linear Time Invariant System ( )tx ( )ty Continuous Linear Time Invariant System ( )tδ ( )th ( ) ( ) 1dtt 0t 0t0 t =δ =∞ ≠ =δ ∫ ∞ ∞− Properties of Delta Function ( )tδ ( )th
  11. 11. Radar Systems Course 11 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Linear Time Invariant Systems Continuous Linear Time Invariant System ( )tx ( )ty Continuous Linear Time Invariant System ( )tδ ( )th Definition : Convolution of Two Functions ( ) ( ) ( ) ( ) ττ−τ≡∗ ∫ ∞ ∞− dtxxtxtx 2121 Reversed and Shifted
  12. 12. Radar Systems Course 12 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Linear Time Invariant Systems Continuous Linear Time Invariant System ( )tx ( )ty Continuous Linear Time Invariant System ( )tδ ( )th ( ) ( ) ( ) ( ) ( ) ττ−τ=∗= ∫ ∞ ∞− dthxthtxty Convolution of and( )tx ( )th • The output of any continuous time, linear, time-invariant (LTI) system is the convolution of the input with the impulse response of the system ( )tx ( )th
  13. 13. Radar Systems Course 13 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Why not Analog Sensors and Calculation Systems ? Voltmeter Torpedo Data Computer (1940s) Slide Rule • Measurement Repeatability • Environmental Sensitivity • Size • Complexity • Cost Disadvantages Courtesy of US Navy Courtesy of Hannes Grobe Courtesy of oschene
  14. 14. Radar Systems Course 14 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society 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
  15. 15. Radar Systems Course 15 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society 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
  16. 16. Radar Systems Course 16 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Waveform Sampling • Sampling converts a continuous signal into a sequence of numbers • Radar signals are complex Continuous-time System ( )tx Discrete-time System [ ]nx A/D Converter
  17. 17. Radar Systems Course 17 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society 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
  18. 18. Radar Systems Course 18 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Ideal Analog to Digital (A/D) Converter INV A/D Converter OUTVINV 12 q2 VERROR =σ OUTV 2 VFS 2 VFS− q)V(P ERROR q 1 2 q 2 q − INOUTERROR VVV −= INV 2 q 2 q −
  19. 19. Radar Systems Course 19 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society “Non-Perfect Nature” of A/D Converters Output Input Offset Actual Ideal • Gain • Missing bits • Monotonicity • Offset • Nonlinearity • Missing bits Input Output Missing Bit Non- Monotonic
  20. 20. Radar Systems Course 20 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Single Tone A/D Converter Testing Frequency (MHz) PowerLevel(dBm) 0 2 4 6 8 -100 -80 -60 -40 -20 0 Fundamental Highest Spur Spur Free Dynamic Range (SFDR) For Ideal A/D S/N=6.02N + 1.76 dB
  21. 21. Radar Systems Course 21 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society A/D Word Length • A / D output is signed N bit integers – Twos complement arithmetic – Quantization noise power = • Signal-to-noise ratio must fit within the word length: – = maximum signal power (target, jamming, clutter) – = thermal noise power in A / D input – Typically, to reduce clipping (limiting) • Required word length: ( ),N/S,SNR o 2 oN 2 S 4≈α SNRlog10SNR 10DB = o 1L N12/1S2 <α>− ( ) 2.16/SNRL DB +> 12/1 A/D Saturation Maximum Signal Noise Quantization Noise Signal Head Room (~10 dB) Foot Room (~10 dB) Receiver Dynamic Range
  22. 22. Radar Systems Course 22 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society 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
  23. 23. Radar Systems Course 23 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Waveform Sampling • Sampling converts a continuous signal into a sequence of numbers • Radar signals are complex Continuous-time System ( )tx Discrete-time System [ ]nx A/D Converter
  24. 24. Radar Systems Course 24 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Sampling - Overview • Sampling Theorem constraint (a.k.a. Nyquist criterion) to prevent “aliasing”: – For continuous aperiodic signals: • Nyquist criterion: – Permits reconstruction via a low pass filtering – Eliminates Aliasing =≥ ss FB2F Sampling Frequency
  25. 25. Radar Systems Course 25 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Signal Sampling Issues • Signal Reconstruction • Elimination of “Aliasing” ( )FX 0 • • • sF2sF LPF ( )FXc 0 B2 B2Fs > ( )FX 0 •• • sF sF3sF2 sF4sF− •• • Overlapping, Aliased Spectra B2Fs <
  26. 26. Radar Systems Course 26 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society The Sampling Theorem • If is strictly band limited, then, may be uniquely recovered from its samples if The frequency is called the Nyquist frequency, and the minimum sampling frequency, , is called the Nyquist rate )t(xc BF0)F(X >= for )t(xc [ ]nx B2 T 2 F S S ≥ π = B2FS = B
  27. 27. Radar Systems Course 27 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Spectrum of a Sampled Signal • Sampling periodically replicates the spectrum – Fourier transform of a sampled signal is periodic • If and are the spectra of and( )FXc ( )FX ( ) ( ) ( ) ( ) ( ) [ ] sF/nF2j n Ft2j n Ft2j cc enx dtenTttgFX dtetxFX π− ∞ −∞= ∞ ∞− π− ∞ −∞= ∞ ∞− π− ∑ ∫ ∑ ∫ = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −δ= = ( )txc [ ]nx ( )FXc 0 ( )FX 0 •• • sF sF3sF2sF− •• •
  28. 28. Radar Systems Course 28 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Distortion of a Signal Spectrum by “Aliasing” • Assume band limited so that 1 B ( )FXc B− F ST/1 B ( )FX B− SFSF− ST/1 2/FS ( )FX SFSF− 2/FS− )t(xc Bf,0)f(X >= • If is sampled with • If is sampled with B2FS ≥ )t(xc )t(xc B2FS < Aliased parts of spectrum for F F No Aliasing ● ● ● ● ● ● ● ● ● ● ● ●
  29. 29. Radar Systems Course 29 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Effect of Sampling Rate on Frequency 0 0 t (sec) 1.0 0.5 - 5 5 0 )t(xc 22c tA c )F2(A A2 )F(X0A,e)t(x π+ =>= − Sampled Signal Its Fourier TransformContinuous Signal Its Fourier Transform [ ] ( ) T 1 F,eaaee)nT(xnx S ATnnATnTA c ====== −−− ( ) [ ] S 2 2 nj n F F 2, acosa21 a1 enxX π=ω +ω− − ==ω ω− ∞ −∞= ∑ ( ) ( ) ( ) ==−= ∑ ∞ −∞= FXˆFFX T 1 FX ccc l l ( ) 2 F FFXT S ≤ 2 F F0 S > )t(xˆc Inverse Fourier TransformReconstructed Signal Adapted from Proakis and Manolakis, Reference 1
  30. 30. Radar Systems Course 30 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Spectrum of Reconstructed Signal Frequency Spectrum ( )FXc Frequency (Hz) Frequency (Hz) Frequency (Hz) Hz3FS = Hz1FS = Signal )t(xc [ ] ( )nTxnx c= [ ] ( )nTxnx c= t (sec) t (sec) t (sec) sec 3 1 T = sec1T = 0 1.0 0.5 1.0 1.0 0.5 0.5 0 0 0 0 0 5 - 5 - 5 - 5 5 5 0 0 0 0 0 0 1 1 1 2 2 2 2 - 2 4 - 2 2 2- 4 - 4 - 2 4 4 - 4 Continuous Signal Sampled Signal Sampled Signal Adapted from Proakis and Manolakis, Reference 1
  31. 31. Radar Systems Course 31 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society 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
  32. 32. Radar Systems Course 32 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Convolution for Discrete Time Systems ( ) ( ) ( ) ττ−τ= ∫ ∞ ∞− dtxhty Continuous Linear Time Invariant System ( )tx ( )ty Continuous-time System ( )tx Discrete-time System [ ]nx Discrete Linear Time Invariant System [ ]ny[ ]nx [ ] [ ] [ ]knxkhny k −= ∑ ∞ −∞=
  33. 33. Radar Systems Course 33 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Graphical Implementation of Convolution [ ] [ ] [ ] [ ] [ ]knhkxknxkhny kk −=−= ∑∑ ∞ −∞= ∞ −∞= Example: 0 1 2 1 2 3 [ ]=kh 1 2 3 4 5 [ ]=kx 2 4 3 11 • Step 1 : Plot the sequences, and[ ]kx [ ]kh
  34. 34. Radar Systems Course 34 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Graphical Implementation of Convolution [ ] [ ] [ ] [ ] [ ]knhkxknxkhny kk −=−= ∑∑ ∞ −∞= ∞ −∞= Example: 0 1 2 1 2 3 [ ]=kh 1 2 3 4 5 [ ]=kx 2 4 3 11 • Step 2 : Take one of the sequences and time reverse it [ ]=− kh -2 -1 0 1 2 3
  35. 35. Radar Systems Course 35 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Graphical Implementation of Convolution [ ] [ ] [ ] [ ] [ ]knhkxknxkhny kk −=−= ∑∑ ∞ −∞= ∞ −∞= Example: 0 1 2 1 2 3 [ ]=kh 1 2 3 4 5 [ ]=kx 2 4 3 11 • Step 3 : Shift by , yielding – a shift to the left – a shift to the right [ ]kh − [ ]=− kh -2 -1 0 1 2 3 n 0n > 0n < [ ]=− knh n-2,n-1,n 1 2 3
  36. 36. Radar Systems Course 36 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Graphical Implementation of Convolution [ ] [ ] [ ] [ ] [ ]knhkxknxkhny kk −=−= ∑∑ ∞ −∞= ∞ −∞= Example: 0 1 2 1 2 3 [ ]=kh 1 2 3 4 5 [ ]=kx 2 4 3 11 • Step 4 : For each value of ,multiply the sequences and ; and add products together for all values of to produce [ ]knh − [ ]=− kh -2 -1 0 1 2 3 n k [ ]kx [ ]ny
  37. 37. Radar Systems Course 37 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Graphical Implementation of Convolution No overlap – = 0 [ ] [ ] [ ] [ ] [ ]knhkxknxkhny kk −=−= ∑∑ ∞ −∞= ∞ −∞= Example: 0 1 2 1 2 3 [ ]=kh [ ]=− kh -2 -1 0 1 2 30nfor = 1 2 3 4 5 [ ]=kx 2 4 3 11 0 1 2 3 4 5 [ ]=kx 2 4 3 11 [ ]=− kh 1 2 3 -2 -1 0 1 2 3 4 5 [ ]ny 10 15 5 0 Outputof Convolution Sample Number [ ]ny 0 1 2 3 4 5 6 7 8 9
  38. 38. Radar Systems Course 38 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Graphical Implementation of Convolution One sample overlaps – = (1x1) = 1 [ ] [ ] [ ] [ ] [ ]knhkxknxkhny kk −=−= ∑∑ ∞ −∞= ∞ −∞= Example: 0 1 2 1 2 3 [ ]=kh [ ]=− k1h -1 0 1 1 2 31nfor = 1 2 3 4 5 [ ]=kx 2 4 3 11 0 1 2 3 4 5 [ ]=kx 2 4 3 11 [ ]=− k1h 1 2 3 -1 0 1 2 3 4 5 [ ]ny 10 15 5 0 Outputof Convolution Sample Number [ ]ny 0 1 2 3 4 5 6 7 8 9
  39. 39. Radar Systems Course 39 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Graphical Implementation of Convolution Two samples overlaps – = (1x2)+(2x1) = 4 [ ] [ ] [ ] [ ] [ ]knhkxknxkhny kk −=−= ∑∑ ∞ −∞= ∞ −∞= Example: 0 1 2 1 2 3 [ ]=kh [ ]=− k2h 0 1 2 1 2 32nfor = 1 2 3 4 5 [ ]=kx 2 4 3 11 0 1 2 3 4 5 [ ]=kx 2 4 3 11 [ ]=− k2h 1 2 3 0 1 2 3 4 5 [ ]ny 10 15 5 0 Outputof Convolution Sample Number [ ]ny 0 1 2 3 4 5 6 7 8 9
  40. 40. Radar Systems Course 40 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Graphical Implementation of Convolution Three samples overlaps – = (1x3)+(2x2)+(4x1) = 11 [ ] [ ] [ ] [ ] [ ]knhkxknxkhny kk −=−= ∑∑ ∞ −∞= ∞ −∞= Example: 0 1 2 1 2 3 [ ]=kh [ ]=− k3h 1 2 33nfor = 1 2 3 4 5 [ ]=kx 2 4 3 11 0 1 2 3 4 5 [ ]=kx 2 4 3 11 [ ]=− k3h 1 2 3 0 1 2 3 4 5 [ ]ny 1 2 3 10 15 5 0 Outputof Convolution Sample Number [ ]ny 0 1 2 3 4 5 6 7 8 9
  41. 41. Radar Systems Course 41 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Graphical Implementation of Convolution Three samples overlaps – = (2x3)+(4x2)+(3x1) = 17 [ ] [ ] [ ] [ ] [ ]knhkxknxkhny kk −=−= ∑∑ ∞ −∞= ∞ −∞= Example: 0 1 2 1 2 3 [ ]=kh [ ]=− k4h 2 3 4 1 2 34nfor = 1 2 3 4 5 [ ]=kx 2 4 3 11 0 1 2 3 4 5 [ ]=kx 2 4 3 11 [ ]=− k4h 1 2 3 0 1 2 3 4 5 6 [ ]ny 10 15 5 0 Outputof Convolution Sample Number [ ]ny 0 1 2 3 4 5 6 7 8 9
  42. 42. Radar Systems Course 42 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Graphical Implementation of Convolution Three samples overlaps – = (4x3)+(3x2)+(1x1) = 17 [ ] [ ] [ ] [ ] [ ]knhkxknxkhny kk −=−= ∑∑ ∞ −∞= ∞ −∞= Example: 0 1 2 1 2 3 [ ]=kh [ ]=− k5h 3 4 5 1 2 35nfor = 1 2 3 4 5 [ ]=kx 2 4 3 11 0 1 2 3 4 5 [ ]=kx 2 4 3 11 [ ]=− k5h 1 2 3 0 1 2 3 4 5 6 [ ]ny 10 15 5 0 Outputof Convolution Sample Number [ ]ny 0 1 2 3 4 5 6 7 8 9
  43. 43. Radar Systems Course 43 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Graphical Implementation of Convolution Two samples overlaps – = (3x3)+(1x2) = 11 [ ] [ ] [ ] [ ] [ ]knhkxknxkhny kk −=−= ∑∑ ∞ −∞= ∞ −∞= Example: 0 1 2 1 2 3 [ ]=kh [ ]=− k6h 1 2 36nfor = 1 2 3 4 5 [ ]=kx 2 4 3 11 0 1 2 3 4 5 [ ]=kx 2 4 3 11 [ ]=− k6h 1 2 3 0 1 2 3 4 5 6 7 [ ]ny 4 5 6 10 15 5 0 Outputof Convolution Sample Number [ ]ny 0 1 2 3 4 5 6 7 8 9
  44. 44. Radar Systems Course 44 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Graphical Implementation of Convolution One sample overlaps – = (1x3) = 3 [ ] [ ] [ ] [ ] [ ]knhkxknxkhny kk −=−= ∑∑ ∞ −∞= ∞ −∞= Example: 0 1 2 1 2 3 [ ]=kh [ ]=− k7h 5 6 7 1 2 37nfor = 1 2 3 4 5 [ ]=kx 2 4 3 11 0 1 2 3 4 5 [ ]=kx 2 4 3 11 [ ]=− k7h 1 2 3 0 1 2 3 4 5 6 7 8 [ ]ny 10 15 5 0 Outputof Convolution Sample Number [ ]ny 0 1 2 3 4 5 6 7 8 9
  45. 45. Radar Systems Course 45 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Graphical Implementation of Convolution No overlap – = 0 [ ] [ ] [ ] [ ] [ ]knhkxknxkhny kk −=−= ∑∑ ∞ −∞= ∞ −∞= Example: 0 1 2 1 2 3 [ ]=kh [ ]=− k8h 6 7 8 1 2 38nfor = 1 2 3 4 5 [ ]=kx 2 4 3 11 0 1 2 3 4 5 [ ]=kx 2 4 3 11 [ ]=− k8h 1 2 3 0 1 2 3 4 5 6 7 8 [ ]ny 10 15 5 0 Outputof Convolution Sample Number [ ]ny 0 1 2 3 4 5 6 7 8 9
  46. 46. Radar Systems Course 46 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Summary- Linear Discrete Time Systems • Any Linear and Time-Invariant (LTI) system can be completely described by its impulse response sequence • The output of any LTI can be determined using the convolution summation • 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 [ ] [ ]nhn H →δ [ ] [ ] [ ] ∞<<∞−−= ∑ ∞ −∞= n,knxkhny k Adapted from MIT LL Lecture Series by D. Manolakis
  47. 47. Radar Systems Course 47 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society 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
  48. 48. Radar Systems Course 48 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society 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
  49. 49. Radar Systems Course 49 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Time and Frequency Domains Analysis Synthesis Fourier Transform Inverse Fourier Transform Time History Frequency Spectrum Frequency DomainTime Domain
  50. 50. Radar Systems Course 50 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society 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
  51. 51. Radar Systems Course 51 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Fourier Transform for Continuous-Time Aperiodic Signals 0 0 Adapted from Manolakis et al, Reference 1 Time Domain Continuous and Aperiodic Signals Frequency Domain Continuous and Aperiodic Signals )t(x )F(X t ∫ ∞ ∞− π− = tde)t(x)F(X tF2j ∫ ∞ ∞− π = dFe)F(X)t(x tF2j F
  52. 52. Radar Systems Course 52 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society 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
  53. 53. Radar Systems Course 53 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Fourier Transform for Discrete-Time Aperiodic Signals Frequency Domain Continuous and Periodic Signals Time Domain Discrete and Aperiodic Signals 4− 2− 20 4 2− π −π 0 π 2πn [ ]nx ω [ ]ωX Adapted from Malolakis et al, Reference 1 [ ] ∫π ω ωω π = 2 nj de)(X 2 1 nx [ ] nj n enX)(X ω− ∞ −∞= ∑=ω
  54. 54. Radar Systems Course 54 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Summary of Time to Frequency Domain Properties 0 1 P F T = 2 ( ) ( ) j Ft X F x t e dt ∞ − π −∞ =∫ 2 ( ) ( ) j Ft x t X F e dF ∞ π −∞ =∫ ( )x t Continuous and Aperiodic Continous and Aperiodic ( )X F Discrete- Time Signals 21 0 1 [ ] N j kn N k n c x n e N π− − = = ∑ 21 0 [ ] N j kn N k k x n c e π− = =∑ [ ]x n kc N− N0 Discrete and Periodic Discrete and Periodic n k Continuous- Time Signals 021 ( ) P j kF t k T P c x t e dt T − π = ∫ 02 ( ) j kF t k k x t c e ∞ π =−∞ = ∑ ( )x t 0 Time-Domain Frequency-Domain Continuous and Periodic Discrete and Aperiodic t 0 kc FPT− PT ( ) [ ] j n n X x n e ∞ − ω =−∞ ω = ∑ 2 1 [ ] ( ) 2 j n x n X e dω π = ω ω π∫ [ ]x n ( )X ω 4− 2− 204 2− π −π 0 π 2π Continous and Periodic n ω Time-Domain Frequency-Domain AperiodicSignals FourierTransforms PeriodicSignals FourierSeries Discrete and Aperiodic 0 t 0 F N− N0 Adapted from Proakis and Manolakis, Reference 1
  55. 55. Radar Systems Course 55 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society 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
  56. 56. Radar Systems Course 56 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Direct DFT Computation • 1. evaluations of trigonometric functions • 2. real ( complex) multiplications • 3. real ( complex) additions • 4. A number of indexing and addressing operations [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]∑ ∑ ∑ − = − = π− − = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π −⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π −= ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π +⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π = =−≤≤= 1N 0n IRI 1N 0n IRR N/nkj2kn N kn N 1N 0n nk N 2 cosnxnk N 2 sinnxkX nk N 2 sinnxnk N 2 cosnxkX eW1Nk0WnxkX 2 N2 2 N4 )1N(N − 2 N )2N(N4 − 2 N≈ Complex MADS MADS Multiply And Divides Adapted from MIT LL Lecture Series by D. Manolakis Aka “Twiddle Factor”
  57. 57. Radar Systems Course 57 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society 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
  58. 58. Radar Systems Course 58 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Fast Fourier Transform (FFT) • An algorithm for each efficiently computing the Discrete Fourier Transform (DFT) and its inverse • DFT MADS (Multiplies and Divides) • FFT MADS • FFT algorithm Development - Cooley / Tukey (1965) Gauss (1805) • 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” ( )2 NO ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Nlog 2 N O 2
  59. 59. Radar Systems Course 59 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Radix 2 Decimation in Time FFT Algorithm • Divide DFT of size N into two interleaved DFTs, each of size N/2 – Example will be – Input to each DFT are even and odd s , respectively • Solve each stage recursively, until the size of the stage’s DFT is 2. [ ] [ ] [ ] N/nkj2kn N kn N 1N 0n N/nkj2 1N 0n eW1Nk0WnxenxkX π− − = π− − = =−≤≤== ∑∑ 82N 3 == [ ]nx [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] kl 2/N 1 2 N 0l k N kl 2/N 1 2 N 0l k)1l2( N 1 2 N 0l kl N 1 2 N 0l kn N Oddn kn N Evenn kn N 1N 0n WlhWWlgWlhWlg WnxWnxWnxkX ∑∑∑∑ ∑∑∑ − = − = + − = − = − = +=+= +== Even index and odd index terms of N/2 point DFT of N/2 point DFT of [ ] [ ]kGlg = [ ]nx [ ] [ ]kHlh =
  60. 60. Radar Systems Course 60 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Radix 2 Decimation in Time FFT Algorithm (continued) • Using the periodicity of the complex exponentials: • And the following properties of the “twiddle factors”: • A block diagram of this computational flow is graphically illustrated in the next chart for an 8 point FFT [ ] [ ] [ ]kHWkGkX kn N+= [ ] [ ] ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ +=⎥ ⎦ ⎤ ⎢ ⎣ ⎡ += 2 N kHkH 2 N kGkG ( )( ) )k(HW2/NkHW WWWW k N )2/N(k N k N 2/N N k N )2/N(k N −=+ −== + + then
  61. 61. Radar Systems Course 61 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society 8 Point Decimation in Time FFT Algorithm (After First Decimation) 4 – Point DFT 4 – Point DFT [ ]0x [ ]1x [ ]2x [ ]3x [ ]4x [ ]5x [ ]6x [ ]7x [ ]0G [ ]1G [ ]2G [ ]3G [ ]0H [ ]1H [ ]2H [ ]3H [ ]0X 0 8W [ ]1X [ ]2X [ ]3X [ ]4X [ ]5X [ ]6X [ ]7X 7 8W 6 8W 5 8W 1 8W 4 8W 2 8W 3 8W
  62. 62. Radar Systems Course 62 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Decimation of 4 Point into two 2 point DFTs • If N/2 is even, and may again be decimated • This leads to: [ ] [ ] [ ] [ ] nk 2/N 1 2 N Oddn nk 2/N 1 2 N Evenn nk 2/N 1 2 N 0n WngWngWngkG ∑∑∑ −−− = +== [ ] [ ] [ ] nk 4/N 1 2 N 0n k 2/N nk 4/N 1 4 N 0n W1n2gWWn2gkG ∑∑ − = − = ++= [ ]ng [ ]nh 2 – Point DFT 2 – Point DFT [ ]0x [ ]2x [ ]4x [ ]6x [ ]0G [ ]1G [ ]2G [ ]3G 0 4W 1 4W 2 4W 3 4W 2 – Point DFT
  63. 63. Radar Systems Course 63 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Butterfly for 2 Point DFT [ ] [ ] [ ]1q0q0Q +=[ ]0q [ ]1q [ ] [ ] [ ]1q0q1Q −= [ ]kq [ ]kQ 1− Now, Putting it all together…..
  64. 64. Radar Systems Course 64 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Flow of 8-Point FFT (Radix 2 - Decimation in Time Algorithm) [ ]0x 0 8W [ ]0X 1 8W 0 8W 0 8W 0 8W 0 8W 0 8W 0 8W 2 8W 2 8W 2 8W 3 8W 1− 1− 1− 1− 1− 1− 1− 1− 1− 1− 1− 1− [ ]4x [ ]2x [ ]6x [ ]1x [ ]5x [ ]3x [ ]7x [ ]1X [ ]2X [ ]3X [ ]4X [ ]5X [ ]6X [ ]7X
  65. 65. Radar Systems Course 65 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Basic FFT Computation Flow Graph • Each “Butterfly” takes 2 MADS (Multiplies and Adds) • Twiddle Factors (For 8 point FFT) • 12 Butterflies implies 12 MADS vs. 64 MADS for 8 point DFT • 512 point FFT more than 100 times faster than 512 DFT ( ) ( ) 2/j1eWjeW 2/j1eeW1eW 4/j33 8 2/j2 8 4/j8/j21 8 00 8 −−==−== −===== π−π− π−π−− N/nkj2nk N eW π− = Check over “Butterfly” “Twiddle” Factor 1− b a r NW bWaA r N+= bWaB r N−= nkr =
  66. 66. Radar Systems Course 66 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Computational Speed – DFT vs. FFT • Discrete Fourier Transform (O ~ N2) • Fast Fourier Transform (O ~ N log2 N) NumberofComplexMultiplications Number of points in Radix 2 FFT Lines Drawn Through Data PointsDFT FFT 104103 102101 101 103 105 107 109 Adapted from Lyons, Reference 2
  67. 67. Radar Systems Course 67 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Fast Fourier Transform (FFT) - Summary • Fast Fourier Transform (FFT) algorithms make possible the computation of DFT with O ((N/2) log2 N) MADS as opposed to O N2 MADS • Many other implementations of the FFT exist: – Radix 2 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. 1, No. 4, pp. 14-21, October 1984
  68. 68. Radar Systems Course 68 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society 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
  69. 69. Radar Systems Course 69 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Finite and Infinite Response Filters • Infinite Impulse Response (IIR) Filters – Output of filter depends on past time history – Example : • Finite Impulse Response (FIR) Filters – Output depends on the finite past – Example: DFT – Other examples: ( )∞− [ ] [ ] [ ]1ny M 1M nx M 1 ny − − += [ ] [ ] N/nkj2 1N 0n enxkX π− − = ∑= [ ] [ ] [ ] [ ] [ ] [ ] [ ]1,nx2,nxnyor 1xnxn,kaky 1N 0n −= = ∑ − =
  70. 70. Radar Systems Course 70 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Four Basic Filter Types- An Idealization Ideal Low Pass Filter Ideal Bandstop FilterIdeal Bandpass Filter Ideal High Pass Filter 1 11 1 ω ω ω ω cω cωcω− cω− π 1ω−1ω− π− 1ω 2ω2ω−2ω− 1ω 2ω π− π−π− ππ π ( )jw eH ( )jw eH ( )jw eH ( )jw eH
  71. 71. Radar Systems Course 71 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society 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
  72. 72. Radar Systems Course 72 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Windowing / Weighting of Filters • If we take a square pulse, sample it M times, and calculate the Fourier transform of this uniform rectangular “window”: • This is recognized as the sinc function which has 13 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 ( ) ( ) ( ) ( ) ( ) ( ) ( ) π≤ω≤π− ω ω =ω ω ω = − − ==ω −− ω− ω−− = ω− ∑ 2/sin 2/Msin W 2/sin 2/Msin e e1 e1 eW 2/1Mj j Mj1M 0n nj
  73. 73. Radar Systems Course 73 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Commonly Used Window Functions • Rectangular • Bartlett (triangular) • Hanning • Hamming • Blackman [ ]=nw [ ]=nw [ ]=nw [ ]=nw [ ]=nw ,0 Mn0,1 ≤≤ otherwise ,0 Mn2/MM/n22 2/Mn0,M/n2 ≤<− ≤≤ otherwise otherwise otherwise ( ) ,0 Mn0,M/n2cos5.05.0 ≤≤π− ( ) ,0 Mn0,M/n2cos46.054.0 ≤≤π− ( ) ( ) ,0 Mn0,M/n4cos08.0M/n2cos5.042.0 ≤≤π+π− otherwise
  74. 74. Radar Systems Course 74 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Comparison of Common Windows Type of Window Peak Sidelobe Amplitude (dB) Approximate Width of Main Lobe Rectangular Bartlett (triangular) Hanning Hamming Blackman ( )1M/4 +π M/8π M/8π M/8π M/12π 13− 57− 31− 25− 41−
  75. 75. Radar Systems Course 75 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Hamming Rectangular Comparison of Rectangular & Hamming Windows 10− 20− 30− 40− 50− 60− 20log10|W(ω| 0 0.1 0.2 0.3 0.4 0.5 Normalized frequency (f = F/Fs)
  76. 76. Radar Systems Course 76 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society 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
  77. 77. Radar Systems Course 77 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society References 1. Proakis, J, G. and Manolakis, D. G., Digital Signal Processing, Principles, Algorithms, and Applications, Prentice Hall, Upper Saddle River, NJ, 4th Ed., 2007 2. Lyons, R. G., Understanding Digital Signal Processing, Prentice Hall, Upper Saddle River, NJ, 2nd Ed., 2004 3. Hsu, H. P., Signals and Systems, McGraw Hill, New York, 1995 4. Hayes, M. H., Digital Signal Processing, McGraw Hill, New York, 1999 5. Oppenheim, A. V. et al, Discrete Time Signal Processing, Prentice Hall, Upper Saddle River, NJ, 2nd Ed., 1999 6. Boulet, B., Fundamentals of Signals and Systems, Prentice Hall, Upper Saddle River, NJ, 2nd Ed., 2000 7. Richards, M. A., Fundamentals of Radar Signal Processing, McGraw Hill, New York, 2005 8. Skolnik, M., Radar Handbook, McGraw Hill, New York, 2nd Ed., 1990 9. Skolnik, M., Introduction to Radar Systems, McGraw Hill, New York, 3rd Ed., 2001
  78. 78. Radar Systems Course 78 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Acknowledgements • Dr Dimitris Manolakis • Dr. Stephen C. Pohlig • Dr William S. Song
  79. 79. Radar Systems Course 79 Review Signals, Systems & DSP 1/1/2010 IEEE New Hampshire Section IEEE AES Society Homework Problems • From Proakis and Manolakis, Reference 1 – Problems 2.1, 2.17, 4.9a and b, 4.10 a and b, 6.1, 6.9 a and b, 8.1 and 8.8 • Or • And from Hays, Reference 4 – Problems 1.41, 1.49, 1.54, 1.59, 2.46, 2.57, 2.58, 3.27, 3.28, 3.34, 6.44, 6.45

×