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Dec 10, 2023 579 likes •704 views

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.

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SLIDE 1

Sampling and Reconstruction

  • f Signals with Finite Rate of

Innovation

Students: Ohad yosef & Tamir Perl Supervisor: Kfir Gedalyahu

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SLIDE 2

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

  • f 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.

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SLIDE 3

Stream of Diracs

Stream of Diracs:

Diracs in an interval of size .

parameters uniquely define the signal, amplitudes and locations.

Local rate of innovation:

1

( ) ( )

k k k

x t x t t 

 

 

2K   

K 2K K K

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SLIDE 4

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.

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SLIDE 5

Results-Unstable reconstruction algorithm

The reconstruction algorithm involves calculating the first moments of using the spanning coefficients and the samples :

Reconstructing stream of six Diracs using B-spline kernel:

High values of the spanning coefficients damage the stability and noise robustness of the reconstruction.

  • 4
  • 2

2 4 6 8 1 2 3 4 5 6 7 8 9

time [sec]

x(t) x(t) reconstruction

  • 15
  • 10
  • 5

5 10 15

  • 5
  • 4
  • 3
  • 2
  • 1

1 2 3 4 5 x 10

11

B-splines Coefficients time [sec] Coefficients

( ) x t

1 ,

[ ] , 0,1,...,

K m m m N n k k n k

c n y a t m N 

  

  

 

, m N

c

n

y

1 N 

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SLIDE 6

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.

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SLIDE 7

Raised Cosine Kernel

Generalization to band limited filter by using Raised Cosine parametric kernel.

A fast decay in the time domain is needed to enable generalization to any stream of Diracs therefore . 1  

5 10 15 20 25 30 35 40 45 50

  • 1
  • 0.5

0.5 1 1.5 2 2.5 3

first “periodic signal” second “periodic signal”

2 2 2

cos ( ) 4 1 t t T h t sinc t T T                

  • 6
  • 4
  • 2
2 4 6
  • 0.2
0.2 0.4 0.6 0.8 1

time [sec] Raised Cosine , time domain =0 =0.25 =0.5 =1

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2
0.2 0.4 0.6 0.8 1
  • 0.2
0.2 0.4 0.6 0.8 1

frequency [Hz] Raised Cosine , frequency domain =0 =0.25 =0.5 =1

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SLIDE 8

Raised Cosine

Unlike ideal LPF Raised Cosine distorts the Fourier coefficients.

Multiplying the coefficients by the inverse filter corrects the distortion.

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.

  • 2
  • 1.5
  • 1
  • 0.5

0.5 1 1.5 2

  • 0.2

0.2 0.4 0.6 0.8 1 1.2 1.4

normalized frequency [fs/2] filters & Coefficients

Raised Cosine Sinc Fourier series coefficients

  • 0.8
  • 0.6
  • 0.4
  • 0.2
0.2 0.4 0.6 0.8 1 2 3 4 5 6 7 8 9 10 11

frequency Correction for Raised Cosine kernel

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SLIDE 9

Denoising methods

B-spline kernel:

Oversampling by factor :

Band limited kernel:

Oversampling.

Cadzow’s iterative approach: M

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SLIDE 10

Simulations

Degradation in performance when using Raised Cosine instead of ideal Sinc.

Using three quarters of the coefficients gives the best results for the Raised Cosine kernel.

5 10 15 20 25 30 35 40 45 50

  • 90
  • 80
  • 70
  • 60
  • 50
  • 40
  • 30
  • 20
  • 10

10

SNR [dB] Locations MSE [dB] Periodic signals comparison 51 samples Sinc Raised Cosine Raised Cosine - half of coefficients Raised Cosine - 1/4 of coefficients Raised Cosine - 3/4 of coefficients

5 10 15 20 25 30 35 40 45 50

  • 90
  • 80
  • 70
  • 60
  • 50
  • 40
  • 30
  • 20
  • 10

10

Local Reconstruction Comparison SNR [dB] Locations MSE [dB] B-Spline Raised Cosine 

For local reconstruction Raised Cosine based algorithm gives better results in reconstructing the same signals than B-spline based algorithm.

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SLIDE 11

Example for “real” implementation

6th order Bessel filter was used as the sampling kernel assuming an effective cutoff frequency:

The input signal consist of short pulses and not of ideal Diracs.

Good recovery of the pulses locations and amplitudes.

R1 50 R2 50 C1 7.2n C3 2.715n C5 1.2732n 1 2 L2 8.85u 1 2 L4 5.08u 1 2 L6 1.08u Vout V2 TD = 2.3u TF = 1p PW = 100p PER = 10u V1 = 0V TR = 1p V2 = 1V V3 TD = 2.8u TF = 1p PW = 100p PER = 10u V1 = 0V TR = 1p V2 = -1.5V

1 2 3 4 5 6 7 8 9 10
  • 1.5
  • 1
  • 0.5
0.5 1 1.5

Reconstruction with Bessel filter Time [sec]

reconstructed signal
  • riginal signal
1 2 3 4 5 6 7 8 9 10
  • 2.5
  • 2
  • 1.5
  • 1
  • 0.5
0.5 1 1.5 x 10
  • 4

Bessel filter output Time [sec]

y(t) yn Frequency 0Hz 0.5MHz 1.0MHz 1.5MHz 2.0MHz 2.5MHz 3.0MHz 3.5MHz 4.0MHz V(VOUT) 0V 200mV 400mV 500mV

Frequency response

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SLIDE 12

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.