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.
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