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Presented ByHarsha Thomas

S2 M.Tech

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Zero-crossing is a commonly used term in electronics,

mathematics, sound, and image processing.

In mathematical terms, a "zero-crossing" is a point

where the sign of a function changes (e.g. from positiveto negative), represented by a crossing of the axis (zero

value) in the graph of the function.

In electronics, the zero-crossing is the instantaneous

point at which there is no voltage present. In a sine wave or other simple waveform, this

normally occurs twice during each cycle.

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Provides an efficient method for reconstructing a band-

limited signal in the discrete domain from its crossings.

The method makes it possible to design A/D converters

that only deliver the crossings, which are then used tointerpolate the input signal at arbitrary instants.

Potentially, it may allow for reductions in power

consumption and complexity in these converters.

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Sampling is the reduction of a continuous signal to a discrete

signal.

Sampler produces samples equivalent to the instantaneous

value of the continuous signal at the desired points.

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Convolution with pulse creates replicas at pulse location

This tells us that the impulse train modulator

Creates images of the Fourier transform of the input signal

Images are periodic with sampling frequency

Ifs< N sampling maybe irreversible due to aliasing of images

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Transmit the digital samples from one point to another, using

digital electronics, rather than analog electronics.

Disadvantages:

Introduces distortion, because we have limited

resolution on the ADC and have limited frequency

of sample rate.

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Audio signal classification. The zero-crossing rate (ZCR) provides a rough estimate of the frequency content

of a signal in time-domain at low computational cost.

Thus, the ZCR has been widely used in many signal processing applications

such as:

1) spectral estimation,

2) voice activity detection,3) voiced/unvoiced speech discrimination,

4) speech recognition,

5) audio sound classification, fluid mechanics,

6) Doppler frequency estimation, and

7) time-delay estimation

The distribution of ZCRs is used for speech and music discrimination. The number of zero-crossings is strongly related with dominant frequency of a

signal.

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Zero crossing based spectrum analyzer: Spectrum analysers are used for measuring the distribution

of signal energy in frequency.

zero-based spectrum analyzer determines the DFT

coefficients of the signal from the zero axis crossings of thesignal rather than from its amplitude samples.

The zero-based approach requires no sampling and

quantization.

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Representation of time-limited bandpass signal by

discrete frequency values.

The locations along the frequency axis at which the

fourier transform of the signal cross zero level.

Sum-of-Sincs model is used.

Signal is reconstructed by solving eigen value problem.

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Key idea is to relate fourier coefficients to zero

crossings of the fourier transform of the signal.

Fourier transform of the signal is computed using filter

banks.

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X(t) has been obtained by windowing a bandpass signal

s(t) using window function w(t).

Fourier transform of x(t):

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Obtain delayed version of x(t).

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The real part of the delayed signal shows more number

of axis crossings .

Number of additional zero-crossings depends on time

delay.

Obtain periodic extension of the signal.

Window the periodic extension by a rectangularwindow whose frequency domain representation

contains sinc functions.

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Determine zero crossings.

Determine fourier coefficients using eigen value

problem.

Signal can be reconstructed using finite Fourier series.

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The analog-to-discrete (A/D) conversion is the first

step for the discrete-time processing of continuous

signals.

This conversion is fundamentally based on the

Sampling Theorem.

Encoding using zero-crossings provide a way to

convert an analog signal into discrete values using SOS

model.

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1) R. Kumaresan , Encoding bandpass signals using zero/level crossings : A model

based approach,IEEE Trans. Speech Audio Process., vol. 18, no. 1, jan 2010.

2) R. Kumaresan and Y. Wang, On the relationship between line spectrum pairs and

zero-crossings of band-pass signals,IEEE Trans. Speech Audio Process., vol. 9,

no. 4, pp. 458461, May 2001.

3) S. M. Kay and R. Sudhaker, A zero crossing-based spectrum analyzer,IEEE

Trans. Acoust., Speech, Signal Process., vol. ASSP-34, no. 1, pp. 96104, Jan.

1986.

4) D. Kim, S. Lee, and R. Kil, Auditory processing of speech signals for robust

speech recognition in real-world noisy environments,IEEE Trans. Speech Audio

Process., vol. 7, no. 1, pp. 5569, Jan. 1999.

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THANK YOU

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Window by a rectangular window

We get

The model for is

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where is the Fourier transform of and * denote

convolution operation. is given by:

where and denote the real and imaginaryparts of respectively and they are frequency

domain sinc functions.

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The real and imaginary parts of is denoted asand , respectively.

Since and are expressed as a linearcombination of shifted versions of the Sinc functions,

the model is called SOS model.

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, k = 1,2,.,p are the locations along the frequency axis atwhich the real part is zero.

, k = 1,2,,q are the locations corresponding to

Writing in matrix-vector notation

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c contains unknown Fourier coefficients.

The submatrices and have p and q rows,

respectively.

c is a unique vector in the null space of X.

Estimate the vector of Fourier coefficients c by

minimizing the quadratic form: .

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The solution vector c is the eigenvector of XTXcorresponding to its smallest eigenvalue.

Fourier transform of delayed signal Fourier

coefficients reconstruct the signal using Fourier

series.

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