# US 2010/0246729 A1 · Sampling method, reconstruction method, and device for sampling and/or reconstructing signals

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- Number: US 2010/0246729 A1
- Document ID: US20100246729A1
- Status: Published application
- Kind: A1
- Application number: 12/814,616
- Filed: 2010-06-14
- Published: 2010-09-30
- Earliest claimed: 2001-03-26
- Assignees: QUALCOMM Incorporated
- Inventors: Martin Vetterli; Pina Marziliano; Thierry Blu
- Web page: https://patentgrep.com/patent/US20100246729A1

## CPC

- H04B 1/707: using direct sequence modulation
- H03M 1/1285: {Synchronous circular sampling, i.e. using undersampling of periodic input signals}
- H03M 1/66: Digital/analogue converters ({H03M1/001 – } H03M1/10 take precedence)
- H04B 2201/70707: Efficiency-related aspects
- H04B: TRANSMISSION
- H03M: CODING; DECODING; CODE CONVERSION IN GENERAL (using fluidic means F15C4/00; optical analogue/digital converters G02F7/00; coding, decoding or code conversion, specially adapted for particular applications, see the relevant subclasses, e.g. G01D, G01R, G06F, G06T, G09G, G10L, G11B, G11C, H04B, H04L, H04M, H04N; ciphering or deciphering for cryptography or other purposes involving the need for secrecy G09C)

## Abstract

Reconstruction method for reconstructing a first signal (x(t)) regularly sampled at a sub-Nyquist rate, comprising the step of retrieving from the regularly spaced sampled values (ys[n], y(nT)) a set of weights (cn, cnr, ck) and shifts (tn, tk) with which said first signal (x(t)) can be reconstructed. The reconstructed signal (x(t)) can be represented as a sequence of known functions (γ(t)) weighted by the weigths (ck) and shifted by the shifts (tk). The sampling rate is at least equal to the rate of innovation (ρ) of the first signal (x(t)).

## Claims

### Claim 1

A method implemented in an apparatus for reconstructing a first signal (x(t)), the method comprising:
sampling a second signal (y(t)) at a sub-Nyquist rate and at uniform intervals;
generating a set of sampled values (ys[n], y(nT)) from the second signal (y(t));
retrieving from said set of sampled values a set of shifts (tn, tk) and weights (cn, cnr, ck); and
reconstructing the first signal (x(t)) based on the set of shifts (tn, tk) and weights (cn, cnr, ck).

### Claim 2 (depends on 1)

Reconstruction method according to claim 1, wherein said set of regularly spaced sampled values comprises at least 2K sampled values (ys[n], y(nT)),
wherein the class of said first signal (x(t)) is known,
wherein the bandwidth (B, |ω|) of said first signal (x(t)) is higher than ωm=π/T, T being the sampling interval,
wherein the rate of innovation (ρ) of said first signal (x(t)) is finite,
wherein said first signal is faithfully reconstructed from said set of sampled values by solving a structured linear system depending on said known class of signal.

### Claim 3 (depends on 1)

Reconstruction method according to claim 1, wherein the reconstructed signal (x(t)) is a faithful representation of the sampled signal (y(t)) or of a signal (xi(t)) related to said sampled signal (y(t)) by a known transfer function (φ(t)).

### Claim 4 (depends on 3)

Reconstruction method according to claim 3, wherein said transfer function (φ(t)) includes the transfer function of a measuring device (7, 9) used for acquiring said second signal (y(t)) and/or of a transfer channel (5) over which said second signal (y(t)) has been transmitted.

### Claim 5 (depends on 1)

Reconstruction method according to claim 1, wherein the reconstructed signal (x(t)) can be represented as a sequence of known functions (γ(t)) weighted by said weights (ck) and shifted by said shifts (tk).

### Claim 6 (depends on 1)

Reconstruction method according to claim 1, wherein the sampling rate is at least equal to the rate of innovation (ρ) of said first signal (x(t)).

### Claim 7 (depends on 1)

Reconstruction method according to claim 1, wherein a first system of equations is solved in order to retrieve said shifts (tk) and a second system of equations is solved in order to retrieve said weights (ck).

### Claim 8 (depends on 7)

Reconstruction method according to claim 7, wherein the Fourier coefficients (X[m]) of said sample values (ys[n]) are computed in order to define the values in said first system of equations.

### Claim 9 (depends on 1)

Reconstruction method according to claim 1, including the following steps:
fording at least 2K spectral values (X[m]) of said first signal (x(t)),
using an annihilating filter for retrieving said arbitrary shifts (tn, tk) from said spectral values (X[m]).

### Claim 10 (depends on 1)

Reconstruction method according to claim 1, wherein said first signal (x(t)) is a periodic signal with a finite rate of innovation (ρ).

### Claim 11 (depends on 10)

Reconstruction method according to claim 10, wherein said first signal (x(t)) is a periodical piecewise polynomial signal, said reconstruction method including the following steps:
finding 2K spectral values (X[m]) of said first signal (x(t)),
using an annihilating filter for finding a differentiated version (xR+1(t)) of said first signal (x(t)) from said spectral values,
integrating said differentiated version to find said first signal.

### Claim 12 (depends on 10)

Reconstruction method according to claim 10, wherein said first signal (x(t)) is a finite stream of weighted Dirac pulses
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,
said reconstruction method including the following steps:
finding the roots of an interpolating filter to find the shifts (tn, tk) of said pulses, solving a linear system to find the weights (amplitude coefficients) (cn, ck) of said pulses.

### Claim 13 (depends on 1)

Reconstruction method according to claim 1, wherein said first signal (x(t)) is a finite length signal with a finite rate of innovation (ρ).

### Claim 14 (depends on 13)

Reconstruction method according to claim 13, wherein said reconstructed signal (x(t)) is related to the sampled signal (y(t)) by a sin c transfer function (φ(t)).

### Claim 15 (depends on 13)

Reconstruction method according to claim 13, wherein said reconstructed signal (x(t)) is related to the sampled signal (y(t)) by a Gaussian transfer function (φσ(t)).

### Claim 16 (depends on 1)

Reconstruction method according to claim 1, wherein said first signal (x(t)) is an infinite length signal in which the rate of innovation (ρ, ρT) is locally finite, said reconstruction method comprising a plurality of successive steps of reconstruction of successive intervals of said first signal (x(t)).

### Claim 17 (depends on 16)

Reconstruction method according to claim 16, wherein said reconstructed signal (x(t)) is related to the sampled signal (y(t)) by a spline transfer function (φ(t)).

### Claim 18 (depends on 16)

Reconstruction method according to claim 16, wherein said first signal (x(t)) is a bilevel signal.

### Claim 19 (depends on 16)

Reconstruction method according to claim 16, wherein said first signal (x(t)) is a bilevel spline signal.

### Claim 20 (depends on 1)

Reconstruction method according to claim 1, wherein said first signal (x(t)) is a CDMA or a Ultra-Wide Band signal.

### Claim 21

A method implemented in an apparatus for sampling a first signal (x(t)), wherein said first signal (x(t)) can be represented over a finite time interval (τ) by the superposition of a finite number (K) of known functions (δ(t), γ(t), γr(t)) delayed by arbitrary shifts (tn, tk) and weighted by arbitrary amplitude coefficients (cn, ck), said method comprising:
convoluting said first signal (x(t)) with a sampling kernel ((φ(t), φ(t)) and using a regular sampling frequency (f, 1/T),
choosing said sampling kernel ((φ(t), φ(t)) and said sampling frequency (f, 1/T) such that sampled values (ys[n], y(nT)) completely specify said first signal (x(t)), and
reconstructing said first signal (x(t)),
wherein said sampling frequency (f, 1/T) is lower than the frequency given by the Shannon theorem, but greater than or equal to twice said finite number (K) divided by said finite time interval (τ).

### Claim 22 (depends on 21)

Sampling method according to claim 21, wherein said first signal (x(t)) is not bandlimited, and wherein said sampling kernel (φ(t)) is chosen so that the number of non-zero sampled values is greater than 2K.