of Signals with Finite Rate of Innovation Students: Ohad yosef - PowerPoint PPT Presentation

Sampling and Reconstruction of Signals with Finite Rate of Innovation Students: Ohad yosef & Tamir Perl Supervisor: Kfir Gedalyahu

Introduction FRI signals are signals which have finite number of degrees of  freedom per unit of time. Rate of innovation is the number of degrees of freedom per unit  of time. The goal is to sample and perfectly reconstruct infinite FRI  signals as close as possible to their rate of innovation. This project will focus on infinite stream of Diracs. 

Stream of Diracs   Stream of Diracs:    x t ( ) x ( t t )  k k  k 1  Diracs in an interval of size . K  parameters uniquely define the signal, amplitudes and K 2 K K  locations. 2 K Local rate of innovation:    

Sampling scheme for infinite length stream of Diracs Sampling scheme:  Reconstruction is done locally by using finite support B-spline  kernel. B-splines are basis functions that can reproduce polynomials.  The goal:  Retrieve the locations & amplitudes from the samples.  Sampling as close as possible to the rate of innovation. 

Results-Unstable reconstruction algorithm The reconstruction algorithm involves calculating the first N  1  moments of using the spanning coefficients and the c x t ( ) m N , samples : y  K 1   n     m c [ ] n y a t , m 0,1,..., N m m N , n k k    n k 0 Reconstructing stream of six Diracs using B-spline kernel:  B-splines Coefficients 11 x 10 9 5 x(t) 8 4 x(t) reconstruction 3 7 2 6 Coefficients 1 5 0 4 -1 3 -2 2 -3 1 -4 0 -5 -4 -2 0 2 4 6 8 -15 -10 -5 0 5 10 15 time [sec] time [sec] High values of the spanning coefficients damage the stability  and noise robustness of the reconstruction.

Problems & suggested solution Problems:  The reconstruction scheme becomes numerically unstable as the number of  Diracs increases. The kernel cannot be realized as an analog filter.  Low noise robustness due to different weights of different samples.  For local reconstruction the sampling rate is higher than the rate of  innovation. Suggested solution:  Extension of the algorithm for periodic stream of Diracs based on ideal LPF  using Fourier series coefficients of the signal (stable reconstruction). Generalization to general band limited kernel.  Generalization to any stream of Diracs using narrow kernel in time domain –  Raised Cosine kernel.

Raised Cosine Kernel Generalization to band limited filter by using Raised Cosine  parametric kernel. Raised Cosine , time domain Raised Cosine , frequency domain  =0  =0  =0.25  =0.25 1 1     =0.5  =0.5 t 0.8  =1  =1   0.8 cos     t T 0.6  0.6 h t sinc   ( )    2 2 0.4 T 4 t 0.4  1 0.2 2 T 0.2 0 0 -0.2 -0.2 -6 -4 -2 0 2 4 6 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 time [sec] frequency [Hz] A fast decay in the time domain is needed to enable  generalization to any stream of Diracs therefore .   1 second 3 first 2.5 “periodic signal” “periodic signal” 2 1.5 1 0.5 0 -0.5 -1 0 5 10 15 20 25 30 35 40 45 50

Raised Cosine Unlike ideal LPF Raised Cosine distorts the Fourier coefficients.  Multiplying the coefficients by the inverse filter corrects the  distortion. filters & Coefficients Correction for Raised Cosine kernel 11 1.4 Raised Cosine 10 Sinc 1.2 Fourier series coefficients 9 8 1 7 0.8 6 0.6 5 4 0.4 3 0.2 2 0 1 0 -0.2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 frequency normalized frequency [f s /2] In the noisy case, multiplying the coefficients by the inverse filter  increases the noise level in the high frequencies: The solution: disposing noisy coefficients at high frequencies. 

Denoising methods B-spline kernel:  Oversampling by factor : M  Band limited kernel:  Oversampling.  Cadzow’s iterative approach: 

Simulations Periodic signals comparison 51 samples Local Reconstruction Comparison 10 10 0 0 -10 -10 -20 -20 Locations MSE [dB] Locations MSE [dB] -30 -30 -40 -40 -50 -50 -60 -60 Sinc -70 Raised Cosine -70 Raised Cosine - half of coefficients B-Spline -80 Raised Cosine - 1/4 of coefficients -80 Raised Cosine Raised Cosine - 3/4 of coefficients -90 0 5 10 15 20 25 30 35 40 45 50 -90 0 5 10 15 20 25 30 35 40 45 50 SNR [dB] SNR [dB] Degradation in performance when For local reconstruction Raised Cosine   using Raised Cosine instead of ideal based algorithm gives better results in Sinc. reconstructing the same signals than B-spline based algorithm. Using three quarters of the coefficients  gives the best results for the Raised Cosine kernel.

Example for “real” implementation 6 th order Bessel filter was used as the sampling kernel assuming  an effective cutoff frequency: 500mV R1 L2 L4 L6 Vout 1 2 1 2 1 2 50 8.85u 5.08u 1.08u 400mV R2 V1 = 0V V2 V2 = 1V C1 C3 C5 50 TD = 2.3u 7.2n 2.715n 1.2732n TR = 1p Frequency response TF = 1p PW = 100p PER = 10u 0 200mV V3 V1 = 0V V2 = -1.5V TD = 2.8u TR = 1p TF = 1p PW = 100p PER = 10u 0V 0Hz 0.5MHz 1.0MHz 1.5MHz 2.0MHz 2.5MHz 3.0MHz 3.5MHz 4.0MHz V(VOUT) Frequency The input signal consist of short pulses and not of ideal Diracs.  Bessel filter output Reconstruction with Bessel filter -4 x 10 1.5 1.5 y(t) reconstructed signal y n original signal 1 1 0.5 0.5 0 -0.5 0 -1 -0.5 -1.5 -1 -2 -2.5 -1.5 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Time [  sec] Time [  sec] Good recovery of the pulses locations and amplitudes. 

Conclusions B-spline based algorithm:  Is numerically unstable, even in the absence of noise.  Obligates sampling at a higher rate than the rate of innovation for local  reconstruction. Cannot be realized as real analog filters.  Has low noise robustness due to different weights of different samples.  Band limited filter based algorithm:  Is stable.  Enables sampling at the rate of innovation.  Gives a better performance in the presence of noise.  Can be realized as real analog filters. 