ECE 421 Introduction to Signal Processing

Size: px

Start display at page:

Download "ECE 421 Introduction to Signal Processing"

Virgil Cain
6 years ago
Views:

1 ECE 421 Introduction to Signal Processing Dror Baron Assistant Professor Dept. of Electrical and Computer Engr. North Carolina State University, NC, USA

2 Denoising and Project 4

3 Where does denoising appear? Setting: y(t)=x a (t)+n(t) Noise n(t) could be Gaussian bell curve noise (thermal noise) Speckle effects saturation sensor returns min/max values Called salt & pepper noise Audio processing record is scratched Large amplitude noise resembling salt & pepper Image processing (Project 4) From 1D signal to 2D image Modern example: In iterative estimation algorithms, n(t) is estimation error Error n(t) becomes smaller over iterations 3

4 How can we denoise? Need to start somewhere exploit structure in input x a (t) Types of structure? Band limited signals signal occupies few freqs w/large coeffs; broadband noise w/small coeffs Knowledge about how much energy (on average!) signal/noise have in different spectral bands More signal barely attenuate More noise attenuate a lot Beyond Fourier coefficients and frequency Piecewise constant signal x a (t) jumps at times < T -1 < T 0 < T 1 < x a (t) t 4

5 Median filters We saw how moving average filters are simple low passes, and can be used to denoise Gaussian noise Noise with outliers (e.g., salt & pepper noise) does not work well with standard low passes Non-linear median filters better for outliers n x(n) Mean Median

6 Project 4 Specifically deals with 2D images and two types of noise Gaussian noise Salt & pepper noise Main concepts in project: From 1D signals to 2D images Linear filters (we ve seen these) good for Gaussian-like noise Non-linear median filters good for salt & pepper For each 3*3 patch surrounding pixel in middle, take median of 9 numbers in patch MMMMMMMMMMMM = 4 Medians less sensitive to outliers! 6

7 Sampling, Reconstruction, and More

8 Roadmap We have seen Chapter 1 from analog to digital and back Chapter 2 discrete time signals & systems; correlation Chapter 3 z-transforms; transfer functions; one-sided z Chapter 4 Fourier transforms Chapter 5 frequency domain analysis of LTI systems About to discuss Chapter 6 Sampling, aliasing, and reconstruction End to end system design Signal denoising (modern topic) Sampling bandpass signals compressed sensing (modern topic) 8

9 Sampling, Aliasing, & Reconstruction [Reading material: Section 6.1]

10 Let s recap We have surveyed 1) Sampling: x(n)=x a (nt) 2) Aliasing: if we sample below Nyquist, higher frequencies get mixed in erroneously 3) Reconstruction: xx aa tt = + nn= xx(nn) ssssss ππ TT (tt nnnn) ππ TT (tt nnnn) Have covered enough math content to discuss in greater detail 10

11 Survey [Personal perspective]

12 Impulse train perspective Have discussed sampling from continuous to discrete time Consider impulse train Δ TT tt = + nn= δδ(tt nnnn) Individual delta δδ(tt nnnn) at time nt Summing over all n yields spacing T between them T (t) 1-3T -2T -T 0 T 2T 3T t 12

13 Fourier perspective on impulse trains Fourier transform of impulse train also impulse train F Δ TT tt + = 1 TT kk= δδ FF kk TT = 1 TT 1 TT FF F{ T (t)} 1/T -2/T -1/T 0 1/T 2/T F Will derive in detail later 13

14 Fourier perspective on impulse trains Fourier transform of impulse train also impulse train F Δ TT tt + = 1 TT kk= δδ FF kk TT = 1 TT 1 TT FF Multiplying x a (t) by Δ TT tt samples input at multiples of T + x a t Δ TT tt = xx aa tt δδ tt nnnn = xx aa (nnnn)δδ tt nnnn nn= + nn= Information in x a (t)δ TT tt identical to that in x(n)=x a (t=nt) 14

15 Fourier perspective on aliasing What s Fourier transform of sampling (product)? F xx aa tt TT tt = F x a (t) F{ TT tt } = X(F) 1 TT 1/TT FF X(F) F{ T (t)} 1/T -B 0 B F * -2/T -1/T 0 1/T 2/T F Recall x(t)*δ(t)=x(t) x(t)*δ(t-nt)=x(t-nt) Delta delayed by nt picks off signal at time nt Sampling: XX FF 1 + TT kk= δδ tt kk TT = 1 TT kk= + XX FF kk TT 15

16 Aliasing -B Sampling convolves X(F) with 1 TT 1 TT(FF) Graphical interpretation: copies of X(F) shifted to left/right by 1/T (and multiplied by 1/T) X(F) 0 B F Three cases: F{ T (t)} 1/T * = -1/T 0 1/T 1) 1/T>2B copies don t overlap no aliasing 2) 1/T<2B copies overlap (our illustration) aliasing 3) 1/T=2B copies touch sounds nice, don t try at home F F{product} -B -1/T 0 B +1/T F 16

17 Active learning Consider signal x a (t) with following Fourier Sketch F{x a (t) T (t)} for three cases a) T=1/2 (sampling at Nyquist) X(F) F b) T=1/5 (sampling above Nyquist) 17

18 Active learning continued c) T=1 (sampling below Nyquist aliasing) X(F) F 18

19 Comments In past, sampling was considered intuitively We understood that sampling sinusoids below 2/cycle was bad New mathematical understanding makes things crisply defined Shifts of Fourier copied left/right Don t want shifts to intersect/overlap 19

20 AM Demodulation Revisited

21 Previous setting Input x a (t) modulated by cosine with noise n(t), z(t)=y(t)+n(t) x a (t) z(t) y(t) cos(f c t) n(t) Spectral domain contains two copies of X(F) with noise everywhere Z(F) -910 KHz -890 KHz +890 KHz +910 KHz F 21

22 Effect of sampling Sampling at 2M samples/sec means spectrum gets copied left/right by 2M Z(F) -900 KHz 900 KHz F KHz 1100 KHz But noise also gets copied more noise than before 22

23 How to reduce noise? Add anti-aliasing filter before sampling Sample at 2M samples/sec filter passes [-1 MHz,+1 MHz] x a (t) y(t) z(t) Anti-aliasing filter Sampler cos(f c t) n(t) Before anti-aliasing filter: broadband noise Sampling would create (aliased) copies of noise After filter: noise limited to [-1 MHz,+1 MHz] Sampling doesn t create copies 23

24 Reconstruction

25 Throwing away shifted copies Recall how sampling copies around shifted versions of X(F) -B X(F) 0 B F F{ T (t)} -1/T 1/T * = 0 1/T Sampling above Nyquist means X(F) lies inside range (-1/2T,+1/2T) F{product} F F{product} -B -1/T 0 B +1/T F -B -1/2T 0 B +1/2T Obtain X(F) by applying low pass filter (LPF) to capture red box F 25

26 What s this LPF? Filtering sampled product by LPF = convolution by sinc xx aa tt = xx aa tt TT tt ssssssss ππ TT tt nnnn = + nn= xx aa (nnnn) ssssssss ππ TT (tt nnnn) 26

27 Example x a (t)=cos(2πf 0 t)=0.5{e +j2πf0t +e -j2πf0t } X(F) Let s sample at F s <2F 0 aliasing Concrete value: F s =1.5F 0 Deltas get shifted right/left by 1.5F 0 X(F) -F 0 +F 0 LPF F Suppose we try to reconstruct -2F 0 -F 0 +F 0-0.5F 0 0.5F 0 +2F 0 F LPF for range (-0.75F 0,+0.75F 0 ) where 0.5F s =0.75F 0 There are erroneous deltas (at ±0.5F 0 ) within this range The real deltas (±F 0 ) aren t there 27

28 Recap Sampling = multiply x a (t) by impulse train T (t) Fourier of product is convolution between X(F) and F{ T (t)} Because F{ T (t)} is also impulse train, X(F) copied around Aliasing = want non-overlapping shifts Sample fast enough no overlap Sample too slow impossible to identify X(F) within product signal Reconstruction = X(F) appears in lower frequencies of product Pick off X(F) using LPF x a (t) Sampler LPF Perfect reconstruction when sampling above Nyquist!! 28

29 Back to the Book (Details)

30 Fourier transform of impulse train Impulse train T (t) is T-periodic can express as Fourier series TT tt = + kk= CC kk ee jjjππππ tt TT Let s compute C k CC kk = 1 TT +TT/2 tt= TT/2 TT (tt)ee jjjππππ TTdddd tt = 1 TT +TT/2 tt= TT/2 δδ(tt)ee jjjππππ tt TTdddd = 1 TT ee jjjππππ0 TT = 1 TT Note: within [-T/2,+T/2] interval there s one delta, which picks off value of exponent at t=0 Fourier series C k can be represented as Fourier transform X(F) with deltas F Δ TT tt = 1 + TT kk= δδ FF kk TT 30

31 Example x a (t)=e -A t Saw in Question 4.2 that X(F)=2A/[A 2 +(2πF) 2 ] Recall signal not band-limited (undefined derivative at t=0) Aliasing no matter how fast we sample However X(F) decays quickly enough that aliasing will be minor 31

32 End-to-End System Design [Reading material: Sections ]

33 Rationale Have seen how sampling / aliasing / reconstruction can be understood via F{xx aa (tt)δ TT (tt)} Will now see how digital signal processing (DSP) can replace analog parts of system Recall that analog parts have various disadvantages Imprecise, noise, non-linearities, Ideally want to use more digital components 33

34 Anti-aliasing filter Signal processing system using DSP: x a (t) Anti-aliasing filter A/D (sampler) DSP D/A y(t) Anti-aliasing filter analog digital analog Sampling introduces copies of X(F) at spacings F s No problem for truly bandlimited X(F) No signal truly bandlimited aliasing reduce with filter 34

35 How to design anti-aliasing filter? Analog LPF has various imperfections H LPF (F) F Practical design introduces various guard bands: #1: bandwidth of desired signal #1 to #2: guard band #2 to #3: transition from pass-band to stop band #3 to #4: guard band #4: beginning of stop band All these require running A/D well above Nyquist 35

36 Digital signal processing Recall end-to-end system x a (t) Anti-aliasing filter A/D (sampler) DSP D/A y(t) + Suppose we want yy aa tt = haa tt= ττ xx aa tt ττ ddττ Corresponds to YY FF = HH aa FF XX aa FF For band-limited x a will focus on HH FF = HH aa FF, FF BB 0, eeeeeeee Cascade of anti-aliasing, A/D, digital filtering with H(F), D/A yields desired output 36

37 Example (+ personal story) In some applications want to delay signal Communications application Symbols being transmitted Symbol = waveform Noise sample at right time periodic sampling A/D samples at wrong time ideal sampling time Solution: estimate non-integer delay, then apply delay filter 37

38 Example continued How does delay filter work? Delay by integer # samples is easy (store in registers) Delay by non-integer samples involves H(ω)=e -jω Need filter structure to depend on 38

39 A/D conversion Have discussed entire signal processing system Will now add details for A/D, quantization, D/A x a (t) Anti-aliasing filter A/D (sampler) DSP D/A y(t) A/D contains two parts Sample & hold analog circuit freezes signal Actual A/D conversion once signal is frozen 39

40 Quantization Digital processing in bits Have finite # bits finite # possible levels for signal Must map continuous valued signal (infinite possible levels) to finite # levels quantization level i-1 quantization level i decision level i-1 decision level i decision level i+1 Partition signal levels into bins B k ={x(n) (level k, level k+1]} x(n) B k assign quantization level k to x(n) 40

41 Structure of bins Quantizers often have uniform step size Level k+1 = Level k + First/last bins extend to ± Advantage uniform quantizers simple to design Disadvantage average error often sub-optimal (same # levels) 41

42 Quantization error Quantization level often in middle of bin Error obeys e q (n)=x(n)-x q (n) (- /2,+ /2] Can show (see book) mean squared error = 2 /12 Requires some assumptions (noise distribution uniform in (- /2,+ /2]) More bins smaller smaller error But more bins also requires more bits Trade-off between bits and error 42

43 D/A conversion How do we convert back to analog? x a (t) Anti-aliasing filter A/D (sampler) DSP D/A y(t) Ideal solution: 1) Modulate impulse train w/digital levels 2) Apply LPF x(n) ideal D/A F s =1/T x a T LPF x a (t) Impossible/impractical to put x(n) on deltas 43

44 D/A using sample & hold Use non-ideal D/A Add sample & hold to stabilize D/A output x(n) D/A sample LPF & hold However sample & hold in time domain convolve x a T by square pulse multiply by sinc in freq domain Can compensate in different ways Ex: Interpolate between samples & run D/A at higher sampling rate No perfect solution, just changes trade-off 44

45 Problem 6.13 Consider continuous time input x a (t) with bandwidth B Input and echo are received together, s a (t)=x a (t)+αx a (t-τ) Applications: Communication signal traverses multiple paths delays/echos Audio same idea (phones use echo cancelation) s a (t) Ideal sampler s(n) H(ω) y(n) Ideal Interpolation y a (t) Challenge: specify F s and H(ω) such that y a (t)=x a (t)? 45

46 Problem 6.13 part 2 What are the ideal components? Ideal sampler = multiply by impulse train Ideal interpolation = assign y(n) to deltas & apply LPF Observation: Fourier(echo)=X(F) exponential Implication: bandwidth(echo)=bandwidth(input)=b F s =2B s(n) contains all information in s a (t) 46

47 Problem 6.13 part 3 Can think of s a (t) and s a (n) as follows: x a (t) G s a (t) x(n) GG s(n) s(n) contains all information about s a (t) apply inverse filter x(n) unknown input GG s(n) GG -1 observed signal x(n) can (try to) recover input This is equivalent to deconvolution We ve discussed how it s quite complicated 47

48 Sampling Bandpass Signals

49 Bandpass signals Recall our old friend the AM modulated signal -910 KHz -890 KHz +890 KHz +910 KHz F Spectral occupancy = = 20 KHz << 910 KHz Quite narrow-band Sampling at Nyquist (2*910=1.82M samples/sec) inefficient Conventional approach in radio systems: Multiply by sinusoidal signal at carrier freq (900 KHz) New signal around DC and around 1.8 MHz Apply LPF to remove copies at ±1.8 MHz New baseband signal can be sampled at 20K samples/sec 49

50 Challenges What if our 900 KHz isn t precise? Copies of original signal moved from ±900 KHz to baseband won t be centered around zero Can be done in principle 50

51 Landau rate Let s go from one pass band to many B 1 T 1 B 2 T 2 B 3 T 3 B 4 T 4 B 5 T 5 B 6 T 6 Each spectral band begins at bottom B i ends at top T i Landau s sampling theorem provides perfect reconstruction for rates above total spectral occupancy Σ i (T i -B i ) Significant expansion over traditional sampling theorem Complicated to implement Must know location of bands, B i and T i Research from 90 s shows that locations need not be known 51

52 Transforms

53 Transform Transforms are (typically) linear operators applied to input vectors x T Y=T(x) Input x R N (N real valued numbers), output y R N Y is linear combination of x, y i =Σ j T ij x j Example Fourier transform of periodic signal in discrete time (finite length N) Weights T ij chosen carefully Want T to be invertible, x=t -1 y Parseval holds (energy conservation) 53

Sparsity Not all transforms are useful Transform useful if energy concentrated in few coefficients This is called sparsity Different classes of signals sparsified by

54 Sparsity Not all transforms are useful Transform useful if energy concentrated in few coefficients This is called sparsity Different classes of signals sparsified by different transforms Example Fourier sparsifies bandpass signals Wavelets for smooth signals w/few discontinuities 2D wavelets (images) most energy in 1-5% of coeffs 54

55 Image compression application Let s apply 2D wavelets to image compression image x 2D wavelets y y q quantization encoding sparse coeffs Output y of transform is very sparse Quantize it with stepsize that takes most coeffs to zero Quantization error e i =y i -y q,i Total energy of error ε=σ i (y i -y q,i ) 2 Error in reconstructed version x, Σ i (x i -x i ) 2 =ε bits y q mostly zero can compress with few bits 55

56 Compressed Sensing [R. Baraniuk, A Lecture in Compressive Sensing. Signal Proc. Mag. 2007]

57 Remaining challenges Transform coding greatly reduce coding rate required to describe signal reduces bandwidth Still need to sample signal Battery operated applications lots of measurements will drain battery High bandwidth applications (radar) required sampling rate pushes limits of A/D s David Donoho: Why go to so much effort to acquire all the data when most of what we get will be thrown away? 57

58 What does compressed sensing do? New framework for sampling, processing, & reconstructing sparse signals Input x R N Only K<<N nonzeros (sparsity) Could be sparse with respect to sparsifying transform T Could have K large coeffs, rest small (still nonzero) Random measurements yy mm = NN nn=1 Only M<<N measurements Elements of Φ mmmm generated randomly M typically somewhat bigger than K Φ mmmm xx nn 58

59 Under determined linear inverse problem Compressed problem can be written in linear algebra form yy 1 yy = Φ xx 11 Φ 12 Φ 1 13 xx 2 Φ 21 Φ 22 Φ 2 23 Matrix contains fewer rows than columns under determined xx 3 General case: No knowledge about x Infinitely many solutions for x impossible to determine x Compressed sensing: x sparse can reconstruct x from few measurements 59

60 Example [Sarvotham, B, & Baraniuk, Sudocodes - Fast Measurement and Reconstruction of Sparse Signals, Int. Symp. Info. Theory, 2006]

61 Example ??????

62 Example What does zero measurement imply? Hint: x strictly sparse ??????

63 Example Graph reduction! ? 0 0???

64 Example What do matching measurements imply? Hint: non-zeros in x are real numbers ? 0 0???

65 Example What is the last entry of x? ?

66 Discussion Compressed sensing can solve under determined linear inverse problem y=φx Compressive signal processing Acquire random linear measurements (in hardware) Use optimization algorithms to search for sparse x that satisfies measurements Described toy problem; can be greatly extended Measurement noise, y=φx+z Input x not sparse but structured Fast reconstruction algorithms Acquisition and processing multiple correlated signals 66

ELEN E4810: Digital Signal Processing Topic 11: Continuous Signals. 1. Sampling and Reconstruction 2. Quantization

ELEN E4810: Digital Signal Processing Topic 11: Continuous Signals 1. Sampling and Reconstruction 2. Quantization 1 1. Sampling & Reconstruction DSP must interact with an analog world: A to D D to A x(t)