Sparse signal representation and the tunable Q-factor wavelet transform

Transcription

1 Sparse signal representation and the tunable Q-factor wavelet transform Ivan Selesnick Polytechnic Institute of New York University Brooklyn, New York

2 Introduction Problem: Decomposition of a signal into the sum of two components:. Oscillatory (rhythmic, tonal) component. Transient (non-oscillatory) component Outline. Signal resonance and Q-factors. Morphological component analysis (MCA) 3. Tunable Q-factor wavelet transform (TQWT) 4. Split augmented Lagrangian shrinkage algorithm (SALSA) 5. Examples References (MDCT, etc) S. N. Levine and J. O. Smith III. Asines+transients+noiseaudiorepresentationfordatacompressionandtime/pitchscale modications. (998) L. Daudet and B. Torrésani. Hybrid representations for audiophonic signal encoding. () S. Molla and B. Torrésani. An hybrid audio coding scheme using hidden Markov models of waveforms. (5) M. E. Davies and L. Daudet. Sparse audio representations using the MCLT. (6)

3 Oscillatory (rhythmic) and Transient Components in EEG Many measured signals have both an oscillatory and a non-oscillatory component. 4 EEG SIGNAL TIME (SECONDS) Rhythms of the EEG: Transients in EEG due to: ) unwanted measurement artifacts Delta Theta Alpha Beta Gamma - 3 Hz 4-7 Hz 8 - Hz - 3 Hz 6 - Hz ) non-rhythmic brain activity (spikes, spindles, and vertex waves) 3

4 Signal resonance and Q-factor PULSE SPECTRUM Q factor = PULSE 4 Q factor = PULSE Q factor = PULSE 4 5 Q factor = TIME (SAMPLES) (a) Signals FREQUENCY (CYCLES/SAMPLE) (b) Spectra. Figure : The resonance of an isolated pulse can be quantified by its Q-factor, defined as the ratio of its center frequency to its bandwidth. Pulses and 3, essentially a single cycle in duration, are low-resonance pulses. Pulses and 4, whose oscillations are more sustained, are high-resonance pulses. 4

5 Resonance-based signal decomposition TEST SIGNAL TEST SIGNAL HIGH RESONANCE COMPONENT LOW PASS FILTERED LOW RESONANCE COMPONENT BAND PASS FILTERED RESIDUAL HIGH PASS FILTERED TIME (SAMPLES) (a) Resonance-based decomposition TIME (SAMPLES) (b) Frequency-based filtering. Figure : Resonance- and frequency-based filtering. (a) Decomposition of a test signal into high- and low-resonance components. The high-resonance signal component is sparsely represented using a high Q-factor WT. Similarly, the low-resonance signal component is sparsely represented using a low Q-factor WT. (b) Decomposition of a test signal into low, mid, and high frequency components using LTI discrete-time filters. 5

6 Resonance-based signal decomposition must be nonlinear SIGNAL LOW RESONANCE COMPONENT = + HIGH RESONANCE COMPONENT = + = + = + = + = + = + Figure 3: Resonance-based signal decomposition must be nonlinear: The signal in the bottom left panel is the sum of the signals above it; however, the low-resonance component of a sum is not the sum of the low-resonance components. The same is true for the high-resonance component. Neither the low- nor high-resonance components satisfy the superposition property. F (s + + s 6 ) 6= F (s )+ + F (s 6 ) 6

7 Rational-dilation wavelet transform (RADWT) Prior work on rational-dilation wavelet transforms addresses the critically-sampled case.. K. Nayebi, T. P. Barnwell III and M. J. T. Smith (99). P. Auscher (99) 3. J. Kovacevic and M. Vetterli (993) 4. T. Blu (993, 996, 998) 5. A. Baussard, F. Nicolier and F. Truchetet (4) 6. G. F. Choueiter and J. R. Glass (7) RADWT (9) gives a solution for the overcomplete case. Reference: Bayram, Selesnick. Frequency-domain design of overcomplete rational-dilation wavelet transforms. IEEE Trans. on Signal Processing, 57, August 9. New: Tunable Q-factor wavelet transform dilation need not be rational. 7

8 Low-pass Scaling x(n) LPS y(n) X(ω) X(ω).5.5 π απ απ π FREQUENCY (ω) π π FREQUENCY (ω) Y(ω) Y(ω).5.5 π π FREQUENCY (ω) π π/α π/α π FREQUENCY (ω) Low-pass scaling with < Low-pass scaling with > 8

9 High-pass Scaling x(n) HPS y(n) X(ω) X(ω).5.5 π ( β)π ( β)π π FREQUENCY (ω) π π FREQUENCY (ω) Y(ω) Y(ω).5.5 π π FREQUENCY (ω) π ( /β)π ( /β)π π FREQUENCY (ω) High-pass scaling with < High-pass scaling with > 9

10 Tunable Q-factor wavelet transform (TQWT) H (!) LPS v (n) LPS / H (!) y (n) x(n) + y(n) H (!) HPS v (n) HPS / H (!) y (n) < apple, < <, + > 8 >< H (!) X(!)! apple Y (!) = >: <! apple 8 ><! < ( ) Y (!) = >: H (!) X(!) ( ) apple! apple 8 H (!) X(!)! < ( ) >< Y (!) = ( H (!) + H (!) ) X(!) ( ) apple! < >: H (!) X(!) apple! apple

11 For perfect reconstruction, the filters should satisfy H (!) =, H (!) =,! apple( ) H (!) =, H (!) =, apple! apple The transition bands of H (!) and of H (!) must be chosen so that H (!) + H (!) = ( ) <! <. () H(ω).5 π απ ( β)π ( β)π απ π FREQUENCY (ω) H(ω).5 π απ ( β)π ( β)π απ π FREQUENCY (ω)

12 The transition band of H (!) and H (!) can be constructed using any power-complementary function, (!), (!)+ (!) =, () We use the Daubechies filter frequency response with two vanishing moments, (!) = ( + cos(!)) p cos(!),! apple. (3) Scale and dilate (!) to obtain transition bands for H (!) and H (!).

13 X(ω).5 π απ ( β)π ( β)π απ π ω (a) Fourier transform of input signal, X(!). H(ω) H(ω).5.5 π απ ( β)π ( β)π απ π FREQUENCY (ω) π απ ( β)π ( β)π απ π FREQUENCY (ω) (b) Frequency responses H (!) and H (!). H(ω) X(ω) H(ω) X(ω).5.5 π απ ( β)π ( β)π απ π ω π απ ( β)π ( β)π απ π ω (c) Fourier transforms of input signal after filtering. V(ω) V(ω).5.5 π.5π.5π π ω π.5π.5π π ω (d) Fourier transforms after scaling. 3

14 Iterated Filters x(n) stage d (n) stage d (n) stage 3 c(n) d 3 (n) Redundancy (total oversampling rate) for many stages: r = When apple, apple, wehavethesystemequivalence: x(n) H (!) LPS H (!) LPS H (!) HPS j stages H (j) (!) LPS j HPS d j (n) 4

15 The equivalent filter is H (j) (!) := 8! < ( ) j >< Yj H (!/ j ) H (!/ m ) ( ) j apple! apple j m= >: j <! apple. (4) H j (!)! ( ) j j Q-factor: Scaling factors, : Q =! c BW = = Q +, = r 5

16 LOW Q-FACTOR WT FREQUENCY RESPONSES 4 LEVELS alpha =.67, beta =., Q =., R = 3. HIGH Q-FACTOR WT FREQUENCY RESPONSES 3 LEVELS alpha =.87, beta =.4, Q = 4., R = π/4 π/ 3π/4 π FREQUENCY (ω)..5 WAVELET TIME (SAMPLES).5. π/4 π/ 3π/4 π FREQUENCY (ω)..5 WAVELET TIME (SAMPLES) Q-factor = Q-factor = 4 Redundancy = 3 Redundancy = 3 6

17 Finite-length Signals... The TWQT can be implemented for finite-length signals using the DFT (FFT)... H (k) LPS N : N v (n) LPS N : N H (k) y (n) x(n) + y(n) H (k) HPS N :N v (n) HPS N :N H (k) y (n) 7

18 Low-pass scaling N :N x(n) LPS N : N y(n) X (k) X (k) N/ N N/ N Y (k) Y (k) N / N N / N Low-pass scaling with N <N Low-pass scaling with N >N. 8

19 High-pass scaling N :N x(n) HPS N :N y(n) X (k) X (k) N/ N N/ N Y (k) Y (k) N / N N / N High-pass scaling with N <N High-pass scaling with N >N 9

20 X (k) N/ N (a) DFT of input signal, X(k). H (k) H (k) T N/ T N T N/ T N (b) Filters H (k) and H (k). The transitions-bands are indicated by T. H (k)x (k) H (k)x (k) N/ N (c) DFT of input signal after filtering. N/ N V (k) V (k) N / N N / N (d) DFT after scaling.

21 Example: Low Q-factor vs High Q-factor WT SUBBANDS (LOW Q RADWT) SUBBANDS (HIGH Q RADWT).89% 7.43% % 6.4% 9 3.5% 3.7%.54% SUBBAND %.8% SUBBAND 7.3% 6.3% 6 6.8% 3 4.7% 4.74% 7 3.9% 5.% 8.38% % 9.% % TIME (SAMPLES) P =, Q = 3, S =, Levels = Dilation =.5, Redundancy = TIME (SAMPLES) P = 5, Q = 6, S =, Levels = Dilation =., Redundancy = 3.

22 Low Q-factor vs High Q-factor WT after sparsification SUBBANDS (LOW Q RADWT) SUBBANDS (HIGH Q RADWT).% 7.% %.7% 9.% %.% SUBBAND %.79% SUBBAND.% 38.66% % 3.% 4.% 7.68% 5 6.6% 8.% % 9.% 7.% TIME (SAMPLES) P =, Q = 3, S =, Levels = Dilation =.5, Redundancy = TIME (SAMPLES) P = 5, Q = 6, S =, Levels = Dilation =., Redundancy = 3.

23 Low Q-factor vs High Q-factor WT SUBBANDS (LOW Q RADWT) SUBBANDS (HIGH Q RADWT) 6.8%.58% 7.75% 6.3% 8 3.% 9 3.8% 3.7% 5.65% SUBBAND 4 6.% SUBBAND 6.8% 6.95% 5.44% 3 8.7% 6.5% 4.37% % 7 3.5% 6.4% % % 8 5.4% 9.4% 9 3.7% TIME (SAMPLES) P =, Q = 3, S =, Levels = Dilation =.5, Redundancy = TIME (SAMPLES) P = 5, Q = 6, S =, Levels = Dilation =., Redundancy = 3. 3

24 Low Q-factor vs High Q-factor WT after sparsification SUBBANDS (LOW Q RADWT) SUBBANDS (HIGH Q RADWT) 6.43%.% % 3.% 8.35% 9.6% 3.99% 4.% SUBBAND % SUBBAND 5.87% 6.5% 5.6% 3 4.% % 4 8.7% 5.7% % % 8.64% 7 9.5% 8.36% 9.3% 9 3.% TIME (SAMPLES) P =, Q = 3, S =, Levels = Dilation =.5, Redundancy = TIME (SAMPLES) P = 5, Q = 6, S =, Levels = Dilation =., Redundancy = 3. 4

25 Constant-Q vs Constant-BW Constant-Q Constant-BW FREQUENCY RESPONSES FREQUENCY RESPONSES FREQUENCY (CYCLES/SAMPLE) FREQUENCY (CYCLES/SAMPLE) ANALYSIS FUNCTION ANALYSIS FUNCTION TIME (SAMPLES) TIME (SAMPLES) ANALYSIS FUNCTION ANALYSIS FUNCTION TIME (SAMPLES) TIME (SAMPLES) fixed resonance frequency-dependent temporal duration frequency-dependent resonance fixed temporal duration 5

26 Tunable Q-factor wavelet transform (TQWT) Run Times Total execution time for forward/inverse N-point TQWT. N time (ms) N time (ms) Run times measured on a base-model Apple MacBook Pro (.4 GHz Intel Core Duo). Cimplementation. 6

27 Tunable Q-factor wavelet transform (TQWT) Summary:. Fully-discrete, modestly overcomplete. Exact perfect reconstruction ( self-inverting ) 3. Adjustable Q-factor: Can attain higher Q-factors than (or same low Q-factor of) the dyadic WT. =) Can achieve higher-frequency resolution needed for oscillatory signals. 4. Samples the time-frequency plane more densely in both time and frequency. =) Exactly invertible, fully-discrete approximation of the continuous WT. 5. FFT-based implementation 7

28 Morphological Component Analysis (MCA) Given an observed signal x = x + x, with x, x, x R N, the goal of MCA is to estimate/determine x and x individually. Assuming that x and x can be sparsely represented in bases (or frames) and respectively, they can be estimated by minimizing the objective function, J(w, w )= kw k + kw k with respect to w and w,subjecttotheconstraint: w + w = x. Then MCA provides the estimates ˆx = w and ˆx = w. Reference: Starck, Elad, Donoho. Image Decomposition via the Combination of Sparse Representations and a Variational Approach, IEEE Trans. on Image Processing, Oct 5. 8

29 Why not a quadratic cost function? If a quadratic cost function is minimized, subject to w + w = x, then, using J(w, w )= kw k + kw k t t = = I, thew and w can be found in closed form: w = w = + + t x, and the estimated components, ˆx = w and ˆx = w,aregivenby Both ˆx and ˆx are just scaled versions of x. =) No separation at all! ˆx = ˆx = + + t x x, x 9

30 MCA as a linear inverse problem The constrained optimization problem can be written as min w,w kw k + kw k subject to w + w = x min w subject to k wk Hw = x where h H = and denotes point-by-point multiplication. 3 i, w = 4 w 5 This is basis pursuit, an `-regularized linear inverse problem... Non-di erentiable Convex =) We use a variant of SALSA (split augmented Lagrangian shrinkage algorithm). Reference: Afonso, Bioucas-Dias, Figueiredo. Fast Image Recovery Using Variable Splitting and Constrained Optimization. IEEE Trans. on Image Processing,. w 3

31 SALSA for MCA (bais pursuit form) Applying SALSA to the MCA problem yields the iterative algorithm: initialize: µ>, d i (5) u i soft(w i + d i,.5 i /µ) d i, i =, (6) c x u u (7) d i t i c i =, (8) w i d i + u i i =, (9) repeat where soft(x, T ) is the soft-threshold rule with threshold T, () soft(x, T )=x max(, T/ x ). Note: no matrix inverses; only forward and inverse transforms. 3

32 9.5 OBJECTIVE FUNCTION ISTA SALSA ITERATION Figure 4: Reduction of objective function during the first iterations. SALSA converges faster than ISTA. 3

33 Example: Resonance-selective nonlinear band-pass filtering (a) TEST SIGNAL (c) OUTPUT OF BAND PASS FILTER 3 4 TIME (SAMPLES) 3 4 TIME (SAMPLES) (b) TWO BAND PASS FILTERS (d) OUTPUT OF BAND PASS FILTER.5 BAND PASS FILTER BAND PASS FILTER FREQUENCY (CYCLES/SAMPLE) 3 4 TIME (SAMPLES) Figure 5: LTI band-pass filtering. The test signal (a) consists of a sinusoidal pulse of frequency. cycles/sample and a transient. Band-pass filters and in (b) are tuned to the frequencies.7 and. cycles/second respectively. The output signals, obtained by filtering the test signal with each of the two band-pass filters, are shown in (c) and (d). The output of band-pass filter, illustrated in (c), contains oscillations due to the transient in the test signal. Moreover, the transient oscillations in (c) have a frequency of.7 Hz even though the test signal (a) contains no sustained oscillatory behavior at this frequency. 33

34 (a) HIGH RESONANCE COMPONENT (c) OUTPUT OF BAND PASS FILTER 3 4 TIME (SAMPLES) 3 4 TIME (SAMPLES) (b) LOW RESONANCE COMPONENT (d) OUTPUT OF BAND PASS FILTER 3 4 TIME (SAMPLES) 3 4 TIME (SAMPLES) Figure 6: Resonance-based decomposition and band-pass filtering. When resonance-based analysis method is applied to the test signal in Fig. 5a, it yields the high- and low-resonance components illustrated in (a) and (b). The output signals, obtained by filtering the high-resonance component (a) with each of the two band-pass filters shown in Fig. 5b, are illustrated in (c) and (d). The transient oscillations in (c) are substantially reduced compared to Fig. 5c. 34

35 Example: Resonance-based decomposition of speech.4 (a) SPEECH SIGNAL [Fs = 6, SAMPLES/SECOND] (b) HIGH RESONANCE COMPONENT (c) LOW RESONANCE COMPONENT TIME (SECONDS) Figure 7: Decomposition of a speech signal ( I m ) into high- and low-resonance components. The high-resonance component (b) contains the sustained oscillations present in the speech signal, while the low-resonance component (c) contains non-oscillatory transients. (The residual is not shown.) 35

36 . (a) ORIGINAL SPEECH (b) HIGH RESONANCE COMPONENT (c) LOW RESONANCE COMPONENT FREQUENCY (Hz) Figure 8: Frequency spectra of the speech signal in Fig. 7 and of the extracted high- and low-resonance components. The spectra are computed using the 5 msec segment from.5 to. seconds. The energy of each resonance component is widely distributed in frequency and their frequency-spectra overlap. 36

37 . (a) RECONSTRUCTION FROM SUBBANDS 9, (b) RECONSTRUCTION FROM SUBBANDS 8, (c) RECONSTRUCTION FROM SUBBANDS (D) SUM OF ABOVE 3 SIGNALS TIME (SECONDS) Figure 9: Frequency decomposition of high-resonance component in Fig. 7. Reconstructing the high-resonance component from a few subbands of the high Q-factor WT at a time, yields an e cient AM/FM decomposition. 37

38 Constant bandwidth + Constant Q-factor CONSTANT BANDWITH FREQUENCY CONSTANT Q FACTOR FREQUENCY AconstantbandwidthandaconstantQ-factordecompositioncanhavehighcoherenceduetosome analysis functions, from each decomposition, having similar frequency support. This can degrade the results of MCA in principle. 38

39 Constant Q-factor: High Q-factor + Low Q-factor CONSTANT Q FACTOR (HIGH Q FACTOR) FREQUENCY CONSTANT Q FACTOR (LOW Q FACTOR) FREQUENCY Two constant Q-factor decompositions with markedly di erent Q-factors will have low coherence because no analysis functions from the two decompositions will have similar frequency support. This is beneficial for the operation of MCA. 39

40 Small coherence between low and high Q-factor WTs p (f) Q /f f /Q p (f) Q /f f /Q f f f Figure : For reliable resonance-based decomposition, the inner product between the low-q and high-q wavelets should be small for all dilations and translations. The computation of the maximum inner product is simplified by assuming the wavelets are ideal band-pass functions and expressing the inner product in the frequency domain. The inner products can be defined in the frequency domain, Z (f,f ):= (f) (f) df, as a function of their center frequencies (equivalently, dilation). The maximum value of the inner product, (f,f ),occurswhenf = f ( + /Q )/( + /Q ) and is given by max = s Q +/ Q +/, Q >Q. () 4

41 Constant Bandwidth: Narrow-band + Wide-band CONSTANT BANDWITH (NARROW BAND) FREQUENCY CONSTANT BANDWITH (WIDE BAND) FREQUENCY Two constant bandwidth decompositions with markedly di erent bandwidths will also have low coherence and are therefore also suitable transform for MCA-based signal decomposition. This gives a bandwidthbased decomposition, rather than a resonance-based decomposition. 4

42 Conclusion: Resonance-based signal decomposition Low Q-factor WT used for sparse representation of the transient component. High Q-factor WT used for sparse representation of the oscillatory (rhythmic) component. Morphological component analysis (MCA) used to separate the two signal components. Oscillatory component not necessarily high-pass contains both low and high frequencies. Transient component not necessarily a low-pass signal contains sharp bumps and jumps. Software available (Matlab software on web, C code by request). 4

43 Graphical User Interface: Facilitates interactive selection and tuning of parameters. 43

44 Group Sparsity with Fully Overlapping Groups Po-Yu Chen Basis pursuit denoising (no group structure): argmin c ky ck + kck Generalized basis pursuit denoising (overlapping group sparsity): argmin c ky ck + NX 3 p c(i) + c(i + ) + c(i + ) i= (group size 3)

45 36 Example: Speech Enhancement FREQUENCY FREQUENCY TIME (SECONDS) Noise-free signal spectrogram TIME (SECONDS) Noisy signal spectrogram 5

46 37 Example: Speech Enhancement (K, K )=(, ) Basis pursuit denoising using ` norm, i.e. no group structure. 7 FREQUENCY FREQUENCY TIME (SECONDS) TIME (SECONDS) 5 The spectrogram exhibits many spurious noise spikes which produces musical noise

47 38 Example: Speech Enhancement (K, K )=(, 8) FREQUENCY FREQUENCY TIME (SECONDS) ! TIME (SECONDS) 5 The spectrogram does not exhibit spurious noise spikes. The denoised signal is free of musical noise artifact

48 39 Example: Speech Enhancement (K, K )=(8, ) FREQUENCY FREQUENCY TIME (SECONDS) ! TIME (SECONDS) 5 The spectrogram does not exhibit spurious noise spikes. The denoised signal is free of musical noise artifact

49 Quasi-Polynomial Approximation via Sparse Derivatives Xiaoran Ning y = x + noise Total variation denoising: argmin x ky xk + kdxk Dual-component polynomial denoising: argmin ky x x k + kdx k + kd x k x,x

50 The clean signal Y: Noisy signal, SNR = 5 db The recovered signal: SNR = 9.5 db

51 Y: Noisy signal, SNR = 5 db.5.5 R : Estimate of X R : Estimate of X The recovered signal = R + R, SNR = db

52 To compare the two methods.5 The clean signal Sparse Derivative Denoising: SNR = db Total Variation Filtering: SNR = 9.5 db Figure: Sparse Derivative Denoising and TV Filtering.

53 . More slides... 44

54 Comparison: EMD & TQWT We compare Empirical Mode Decomposition (EMD) Sparse TQWT Representation A goal of EMD is to decompose a multicomponent signal into several narrow-band components (intrinsic mode functions). For some real signals, a sparse TQWT representation leads to a more reasonable decomposition than EMD (next two slides). 45

55 7 Empirical mode decomposition (EMD) IMF index Time (seconds) 46

56 Sparse TQWT Representation band band band 3 band 4 residual recon Time (seconds) 47

57 Morphological Component Analysis (MCA) Noisy signal case In the noisy case, we should not ask for exact equality. Given an observed signal x = x + x + n, with x, x, x R N, where n is noise, the components x and x can be estimated by minimizing the objective function, J(w, w )=kx w w k + kw k + kw k with respect to w and w.thenmcaprovidestheestimates ˆx = w and ˆx = w. Reference: Starck, Elad, Donoho. Image Decomposition via the Combination of Sparse Representations and a Variational Approach, IEEE Trans. on Image Processing, Oct 5. 48

58 Why not a quadratic cost function? If the `-norm is used for the penalty term, J(w, w )=kx w w k + kw k + kw k, then, using t t = = I, theminimizingw and w can be found in closed form: w = + + t x t w = + + x and the estimated components, ˆx = w and ˆx = w,aregivenby ˆx = ˆx = Both ˆx and ˆx are just scaled versions of x. =) No separation at all! + + x + + x 49

59 MCA as a linear inverse problem The objective function can be written as J(w, w )=kx w w k + kw k + kw k J(w) =kx Hwk + k wk where H = h i, w = 3 4 w 5 w and denotes point-by-point multiplication. An `-regularized linear inverse problem... Non-di erentiable Convex =) Use Iterative Soft Thresholding Algorithm (ISTA) or another algorithm to minimize J(w). Other algorithms include SparSA, TwIST, FISTA, SALSA, etc. 5

60 Split augmented Lagrangian shrinkage algorithm (SALSA) SALSA is an algorithm for minimizing SALSA is based on the minimization of J(w) =kx Hwk + kwk by the alternating split augmented Lagrangian algorithm: min u f (u)+f (u) () u (k+) =argmin u f (u)+µku v (k) d (k) k (3) v (k+) =argmin v f (v)+µku (k+) v d (k) k (4) d (k+) = d (k) u (k+) + v (k+) (5) Reference: Afonso, Bioucas-Dias, Figueiredo. Fast Image Recovery Using Variable Splitting and Constrained Optimization. IEEE Trans. on Image Processing,. 5

61 SALSA Applying SALSA to the MCA problem yields the iterative algorithm: initialize: µ>, d (6) u i soft(w i + d i,.5 i /µ) d i, i =, (7) c x u u (8) t d i i c i =, (9) µ + w i d i + u i, i =, () repeat where soft(x, T ) is the soft-threshold rule with threshold T, () soft(x, T )=x max(, T/ x ). Note: no matrix inverses; only forward and inverse transforms. 5

62 NOISY SPEECH WAVEFORM HIGH Q FACTOR COMPONENT LOW Q FACTOR COMPONENT RESIDUAL TIME (SECONDS) 53