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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
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1
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|
2
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 ]