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Lecture 12: Parametric Signal ModelingXILIANG LUO2014/11

Discrete Time Signals Mathematically, discrete-time signals can be expressed as a sequence of numbers

In practice, we obtain a discrete-time signal by sampling a continuous-time signal as:

where T is the sampling period and the sampling frequency is defined as 1/T

Representation of Sequences by FT Many sequences can be represented by a Fourier integral as follows:

x[n] can be represented as a superposition of infinitesimally small complex exponentials

Fourier transform is to determine how much of each frequency component is used to synthesize the sequence

𝑥 [𝑛 ]= 12𝜋 ∫

−𝜋

𝜋

𝑋 (𝑒 𝑗 𝜔 )𝑒 𝑗𝜔𝑛𝑑𝜔

𝑋 (𝑒 𝑗𝜔 )=∑𝑛

❑

𝑥[𝑛]𝑒− 𝑗𝜔𝑛

Z-Transform

a function of the complex variable: z

If we replace the complex variable z by , we have the Fourier Transform!

Periodic Sequence Discrete Fourier Series

For a sequence with period N, we only need N DFS coefs

Discrete Fourier Transform

DFT is just sampling the unit-circle of the DTFT of x[n]

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Parametric Signal Modeling

A signal is represented by a mathematical model which has a Predefined structure involving a limited number of parameters.

A given signal is represented by choosing the specific set ofparameters that results in the model output being as closeas possible in some prescribed sense to the given signal.

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Parametric Signal Modeling

LTIH(z)

v[n] s’[n]

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All-Pole Modeling𝐻 (𝑧 )= 𝐺

1−∑𝑘=1

𝑝

𝑎𝑘 𝑧−𝑘

�̂� [𝑛 ]=∑𝑘=1

𝑝

𝑎𝑘 �̂� [𝑛−𝑘 ]+𝐺𝑣 [𝑛]

All-pole model assumes the signal can be approximated as a linearcombination of its previous values!

this modeling is also called: linear prediction

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All-Pole Modeling Least Squares Approximation

�̂� [𝑛 ]=∑𝑘=1

𝑝

𝑎𝑘 �̂� [𝑛−𝑘 ]+𝐺𝑣 [𝑛]

min ∑𝑛=−∞

∞

(𝑠 [𝑛 ]− �̂�[𝑛] )2

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All-Pole Modeling Least Squares Inverse Model

𝐻 (𝑧 )= 𝐺

1−∑𝑘=1

𝑝

𝑎𝑘 𝑧−𝑘 𝐴 (𝑧 )=1−∑

𝑘=1

𝑝

𝑎𝑘𝑧−𝑘

LTIA(z)

s[n] g[n]

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All-Pole Modeling Least Squares Inverse Model

LTIA(z)

s[n] g[n]

𝑔 [𝑛 ]=𝑠 [𝑛 ]−∑𝑘=1

𝑝

𝑎𝑘𝑠 [𝑛−𝑘 ]

�̂� [𝑛 ]=𝑔 [𝑛 ]−𝐺𝑣 [𝑛]

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All-Pole Modeling Least Squares Inverse Model

min ℰ=⟨|�̂� [𝑛 ]|2 ⟩

�̂� [𝑛 ]=𝑔 [𝑛]−𝐺𝑣 [𝑛]

ℰ=⟨𝑒2[𝑛 ]⟩+𝐺2 ⟨𝑣2[𝑛] ⟩−2G ⟨𝑣 [𝑛 ]𝑒[𝑛] ⟩

∑𝑘=1

𝑝

𝑎𝑘 ⟨𝑠 [𝑛−𝑖 ] 𝑠[𝑛−𝑘] ⟩=⟨𝑠 [𝑛− 𝑖 ] 𝑠 [𝑛]⟩

Yule-Walker equations

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Linear Predictor𝑒 [𝑛 ]=𝑠 [𝑛 ]−∑

𝑘=1

𝑝

𝑎𝑘𝑠 [𝑛−𝑘]

1. if input v[n] is impulse, the prediction error is zero2. if input v[n] is white, the prediction error is white

Linear Predictor

+s[n]

s’[n]

e[n]

-

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Deterministic Signal

Minimize total error energy will render the following definitions:

min ℰ=⟨|�̂� [𝑛 ]|2 ⟩

∑𝑘=1

𝑝

𝑎𝑘 ⟨𝑠 [𝑛−𝑖 ] 𝑠[𝑛−𝑘] ⟩=⟨𝑠 [𝑛− 𝑖 ] 𝑠 [𝑛]⟩

𝜙𝑠𝑠 [𝑖 ,𝑘 ] :=∑𝑛

❑

𝑠 [𝑛 ] ¿¿

∑𝑘=1

𝑝

𝑎𝑘𝑟𝑠𝑠 [ 𝑖−𝑘 ]=𝑟𝑠𝑠 [𝑖 ]

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Random Signal

Minimize expected error energy will render the following definitions:

min ℰ=⟨|�̂� [𝑛 ]|2 ⟩

∑𝑘=1

𝑝

𝑎𝑘 ⟨𝑠 [𝑛−𝑖 ] 𝑠[𝑛−𝑘] ⟩=⟨𝑠 [𝑛− 𝑖 ] 𝑠 [𝑛]⟩

𝜙𝑠𝑠 [𝑖 ,𝑘 ]≔𝐸 {𝑠 [𝑛 ]¿

∑𝑘=1

𝑝

𝑎𝑘𝑟𝑠𝑠 [ 𝑖−𝑘 ]=𝑟𝑠𝑠 [𝑖 ]

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All-Pole SpectrumAll-pole method gives a method of obtaining high-resolution estimates ofa signal’s spectrum from truncated or windowed data!

Spectrum Estimate ¿| 𝐺

1−∑𝑘=1

𝑝

𝑎𝑘𝑒− 𝑗 𝜔𝑘|

2

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All-Pole SpectrumSpectrum Estimate ¿| 𝐺

1−∑𝑘=1

𝑝

𝑎𝑘𝑒− 𝑗 𝜔𝑘|

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For deterministic signal, we have the following DTFT:

𝑆 (𝑒 𝑗 𝜔)=∑𝑛=0

𝑀

𝑠 [𝑛 ]𝑒− 𝑗𝜔𝑛

𝑟 𝑠𝑠 [𝑚 ]= ∑𝑛=0

𝑀−∨𝑚∨¿𝑠 [𝑛 ] 𝑠¿ ¿¿

¿

𝑅𝑠𝑠 (𝑒 𝑗 𝜔 )=|𝑆 (𝑒 𝑗𝜔 )|2

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All-Pole Analysis of Speech

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Solution to Yule-Walker Eq.

Φ 𝑎=Ψ

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Solution to Yule-Walker Eq.

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All-Zero ModelMoving-Average Model:

𝑢 (𝑛)=𝑣 [𝑛 ]+∑𝑘=1

𝐾

𝑏𝑘𝑣 [𝑛−𝑘]

𝐻 (𝑧 )=1+∑𝑘=1

𝐾

𝑏𝑘𝑧−𝑘

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ARMA Model𝑢 (𝑛)+∑

𝑘=1

𝑀

𝑎𝑘𝑢[𝑛−𝑚]=𝑣 [𝑛 ]+∑𝑘=1

𝐾

𝑏𝑘𝑣 [𝑛−𝑘]

𝐻 (𝑧 )=1+∑

𝑘=1

𝐾

𝑏𝑘𝑧−𝑘

1+∑𝑘=1

𝑀

𝑎𝑘𝑧−𝑘

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Wold DecompositionWold (1938) proved a fundamental theorem: any stationary discretetime stochastic process may be decomposed into the sum of a generallinear process and a predictable process, with these two processesbeing uncorrelated with each other.

𝑥 [𝑛 ]=𝑢[𝑛]+𝑠 [𝑛]

𝑢 [𝑛 ]=∑𝑘=0

∞

𝑏𝑘𝑣 [𝑛−𝑘]

𝑣 [𝑛 ]=∑𝑘=1

∞

𝑎𝑘𝑣 [𝑛−𝑘 ]