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Part 2 discusses power spectrum analysis. Part 4 examines amplitude and frequency modulation. It
will be published Thursday, March 13.
5.2.4 Parametric spectrum analysis
Non-parametric spectral density estimation methods like the Fourier analysis described above are well studied and established. Unfortunately, this class of methods has some drawbacks. When the available set of N samples is small, resolution in frequency is severely limited. Also auto-correlation outside the sample set is considered to be zero, i.e. Rxx(k) = 0 for k > N , which may be an unrealistic assumption in many cases.
Fourier-based methods assume that data outside the observed window is either periodic or zero. The estimate is not only an estimate of the observed N data samples, but also an estimate of the "unknown" data samples outside the windows, under the assumptions above. Alternative estimation methods can be found in the class of model-based , parametric spectrum analysis methods (Mitra and Kaiser, 1993). Some of the most common methods will be presented briefly below.
The underlying idea of assuming the signal to be analyzed can be generated using a model filter. The filter has the causal transfer function H(z) and a white noise input signal e(n), with mean value zero and variance σe2. In this case, the output signal x(n) from the filter is a wide-sense stationary signal with power density
Hence, the power density can be obtained if the model filter transfer function H(z) is known, i.e. if the model type and its associated filter parameters are known. The parametric spectrum analysis can be divided into three steps:
- Selecting an appropriate model and selecting the order of H(z). There are often many different possibilities
- Estimating the filter coefficients, i.e. the parameters from the N data samples x(n) where n = 0, 1, ..., N − 1
- Evaluating the power density, as in equation (5.40) above.
Selecting a model is often easier if some prior knowledge of the signal is available. Different models may give more or less accurate results, but may also be more or less computationally demanding. A common model type is the auto-regressive moving average (ARMA) model
where h(n) is the impulse response of the filter and the denominator polynomial A(z) has all its roots inside the unit circle, for the filter to be stable. The ARMA model may be simplified. If q = 0, then B(z) = 1 and an all-pole filter results, yielding the auto-regressive (AR) model
or, if p = 0, then A(z) = 1 and a filter having only zeros results, a so-called moving average (MA) model
Note, these models are mainly infinite impulse response (IIR) and finite impulse response (FIR) filters (see also Chapter 1). Once a reasonable model is chosen, the next measure is estimating the filter parameters based on the available data samples. This can be done in a number of ways. One way is by using the auto-correlation properties of the data sequence x(n). Assuming an ARMA model as in equation (5.41), the corresponding difference equation (see Chapter 1) can be written as
where e(n) is the white noise input sequence. Multiplying equation (5.44) by x*(n + m) and taking the expected value we obtain
where the auto-correlation of x(n) is
and the cross-correlation between x(n) and e(n) is
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