Lecture 12: Parametric Signal Modeling

XILIANG LUO 2014/11 1

Discrete Time Signals  Mathematically, discrete-time signals can be expressed as a sequence of numbers 𝑥 𝑛 , 𝑛 ∈ 𝑍  In practice, we obtain a discrete-time signal by sampling a continuous time signal as: 𝑥 𝑛 = 𝑥 𝑎 (𝑛𝑇) 1/T where T is the sampling period and the sampling frequency is defined as

Representation of Sequences by FT  Many sequences can be represented by a Fourier integral as follows: 𝑥 𝑛 = 1 𝜋 2𝜋 −𝜋 𝑋 𝑒 𝑗𝜔 𝑒 𝑗𝜔𝑛 𝑑𝜔 𝑋 𝑒 𝑗𝜔 = 𝑥[𝑛] 𝑒 −𝑗𝜔𝑛 𝑛  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

Z-Transform a function of the complex variable: z If we replace the complex variable z by 𝑒 𝑗𝜔 , 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]

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 of parameters that results in the model output being as close as possible in some prescribed sense to the given signal.

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Parametric Signal Modeling s’[n] v[n] LTI H(z) 8

All-Pole Modeling 𝐻 𝑧 = 𝐺 1 − 𝑝 𝑘=1 𝑎 𝑘 𝑧 −𝑘 𝑝 𝑠 𝑛 = 𝑎 𝑘 𝑘=1 𝑠[𝑛 − 𝑘] + 𝐺𝑣[𝑛] All-pole model assumes the signal can be approximated as a linear combination of its previous values!

 this modeling is also called:

linear prediction

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All-Pole Modeling  Least Squares Approximation 𝑝 𝑠 𝑛 = 𝑎 𝑘 𝑘=1 𝑠[𝑛 − 𝑘] + 𝐺𝑣[𝑛] ∞ min 𝑛=−∞ 𝑠 𝑛 − 𝑠[𝑛] 2 10

All-Pole Modeling  Least Squares Inverse Model 𝐻 𝑧 = 𝐺 1 − 𝑝 𝑘=1 𝑎 𝑘 𝑧 −𝑘 𝑝 𝐴 𝑧 = 1 − 𝑎 𝑘 𝑧 −𝑘 𝑘=1 g[n] s[n] LTI A(z) 11

All-Pole Modeling  Least Squares Inverse Model s[n] LTI A(z) 𝑝 𝑔 𝑛 = 𝑠 𝑛 − 𝑎 𝑘 𝑠 [𝑛 − 𝑘] 𝑘=1 𝑒 𝑛 = 𝑔 𝑛 − 𝐺𝑣[𝑛] g[n] 12

All-Pole Modeling  Least Squares Inverse Model 𝑒 𝑛 = 𝑔 𝑛 − 𝐺𝑣[𝑛] min ℰ = 𝑒 𝑛 2 ℰ = 𝑒 2 [𝑛] + 𝐺 2 𝑣 2 [𝑛] − 2G 𝑣 𝑛 𝑒[𝑛] 𝑝 𝑎 𝑘 𝑘=1 𝑠 𝑛 − 𝑖 𝑠[𝑛 − 𝑘] = 𝑠 𝑛 − 𝑖 𝑠[𝑛] Yule-Walker equations 13

Linear Predictor 𝑝 𝑒 𝑛 = 𝑠 𝑛 − 𝑎 𝑘 𝑠 [𝑛 − 𝑘] 𝑘=1 1.

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if input v[n] is impulse, the prediction error is zero if input v[n] is white, the prediction error is white s[n] Linear Predictor + e[n] s’[n] 14

Deterministic Signal min ℰ = 𝑒 𝑛 2 𝑝 𝑎 𝑘 𝑘=1 𝑠 𝑛 − 𝑖 𝑠[𝑛 − 𝑘] = 𝑠 𝑛 − 𝑖 𝑠[𝑛] Minimize total error energy will render the following definitions: 𝜙 𝑠𝑠 𝑖, 𝑘 : = 𝑠 𝑛 [𝑠[𝑛 − 𝑖 − 𝑘 ] = 𝑟 𝑠𝑠 [𝑖 − 𝑘] 𝑛 𝑝 𝑎 𝑘 𝑟 𝑠𝑠 𝑘=1 𝑖 − 𝑘 = 𝑟 𝑠𝑠 [𝑖] 15

Random Signal min ℰ = 𝑒 𝑛 2 𝑝 𝑎 𝑘 𝑘=1 𝑠 𝑛 − 𝑖 𝑠[𝑛 − 𝑘] = 𝑠 𝑛 − 𝑖 𝑠[𝑛] Minimize expected error energy will render the following definitions: 𝜙 𝑠𝑠 𝑖, 𝑘 ≔ 𝐸{𝑠 𝑛 [𝑠[𝑛 − 𝑖 − 𝑘 ]} = 𝑟 𝑠𝑠 [𝑖 − 𝑘] 𝑝 𝑎 𝑘 𝑟 𝑠𝑠 𝑘=1 𝑖 − 𝑘 = 𝑟 𝑠𝑠 [𝑖] 16

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

Spectrum Estimate = 𝐺 1 − 𝑝 𝑘=1 𝑎 𝑘 𝑒 −𝑗𝜔𝑘 2 17

All-Pole Spectrum Spectrum Estimate = 𝐺 1 − 𝑝 𝑘=1 𝑎 𝑘 𝑒 −𝑗𝜔𝑘 2 For deterministic signal, we have the following DTFT: 𝑀 𝑆(𝑒 𝑗𝜔 ) = 𝑠 𝑛 𝑒 −𝑗𝜔𝑛 𝑛=0 𝑀−|𝑚| 𝑟 𝑠𝑠 𝑚 = 𝑛=0 𝑠 𝑛 𝑠[𝑛 + |𝑚|] 𝑅 𝑠𝑠 𝑒 𝑗𝜔 = 𝑆 𝑒 𝑗𝜔 2 18

All-Pole Analysis of Speech 19

Solution to Yule-Walker Eq.

Φ𝑎 = Ψ 20

Solution to Yule-Walker Eq.

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All-Zero Model Moving-Average Model: 𝐾 𝑢 𝑛 = 𝑣 𝑛 + 𝑏 𝑘 𝑣[𝑛 − 𝑘] 𝑘=1 𝐾 𝐻 𝑧 = 1 + 𝑏 𝑘 𝑧 −𝑘 𝑘=1 22

ARMA Model 𝑀 𝐾 𝑢 𝑛 + 𝑎 𝑘 𝑢[𝑛 − 𝑚] = 𝑣 𝑛 + 𝑏 𝑘 𝑣[𝑛 − 𝑘] 𝑘=1 𝑘=1 𝐻 𝑧 = 1 + 𝐾 𝑘=1 1 + 𝑀 𝑘=1 𝑏 𝑘 𝑧 −𝑘 𝑎 𝑘 𝑧 −𝑘 23

Wold Decomposition Wold (1938) proved a fundamental theorem: any stationary discrete time stochastic process may be decomposed into the sum of a general linear process and a predictable process, with these two processes being uncorrelated with each other.

𝑥 𝑛 = 𝑢[𝑛] + 𝑠[𝑛] ∞ 𝑢 𝑛 = 𝑏 𝑘 𝑣[𝑛 − 𝑘] 𝑘=0 ∞ 𝑣 𝑛 = 𝑎 𝑘 𝑣[𝑛 − 𝑘] 𝑘=1 24