Spectra Averaging

Defining the time series,

Figure 1. EQUATION_DISPLAY

{h (x)} = {h (x; t), t \in {T}}

(521)

Figure 2. EQUATION_DISPLAY

{T} = {t_{0}, \dots, t_{N - 1}}

(522)

One method of temporal spectral averaging is considered here: blocking.

Blocking is the process of smoothing a spectra. The initial signal is broken into a number of subsignals of the same length. At the same time, sampling granularity is held constant.

Figure 3. EQUATION_DISPLAY

{h x} = {{h_{j}^{b} (x)}, j = 1, N_{b}}

(523)

{h_{j}^{b} (x)} = {h (x; t), t \in {T_{j}}}

(524)

Figure 4. EQUATION_DISPLAY

{T_{j}} = t_{\frac{j N}{N_{b}}}, t_{\frac{j N}{N_{b}} + 1}, \dots, t_{\frac{(j + 1) N}{N_{b}} - 1}

(525)

Additional blocks can be overlapped using an overlap factor, which can have values in the range $0 < α < 0.9$ . In this case, the number of blocks is

Figure 5. EQUATION_DISPLAY

N_{b, α} = N_{b} + (N_{b} - 1) [\frac{α}{1 - α}]

(526)

and they are defined as

Figure 6. EQUATION_DISPLAY

{h_{j, α}^{b} (x)} = {h (x; t), t \in {T_{j}}}

(527)

Figure 7. EQUATION_DISPLAY

{T_{j}} = t_{\frac{j + (1 - α) N}{N_{b}}}, t_{\frac{j + (1 - α) N}{N_{b}} + 1}, \dots, t_{\frac{(j + 1) N}{N_{b}} - 1}

(528)

Then, treating each subsignal as an independent signal, define the blocking periods

Figure 8. EQUATION_DISPLAY

{T_{b}} = {t_{0}, t_{1}, \dots, t_{\frac{N}{N_{b}} - 1}}

(529)

over which the mean time history due to blocking can be calculated.

Figure 9. EQUATION_DISPLAY

{{\bar{h}}^{b} (x)} = \frac{1}{N_{b, α}} \sum_{j = N_{b}}^{} {h_{j, α}^{b} (x)}

(530)

Defining a temporal mean spectrum due to the mean of the Fourier transforms of a set of subsignals,

Figure 10. EQUATION_DISPLAY

{\bar{g}}_{t} (x; ω) = \frac{1}{N} \sum_{j = 1, N}^{} g_{j} (x, ω)

(531)

From the Fourier transform addition theorem, you can see that:

Figure 11. EQUATION_DISPLAY

F T [\bar{h} (x; t)] = F T [\frac{1}{M} \sum_{j = 1, M}^{} h_{j} (x; t)] = \frac{1}{M} \sum_{j = 1, M}^{} F T [h_{j} (x; t)]

(532)

Figure 12. EQUATION_DISPLAY

{\bar{g}}_{t} (x; ω) = F T [\bar{h} (x; t)]

(533)

Blocking reduces the length of time series. As a result, low frequencies are less well characterized.

It is also possible to average spectra spatially. The spatial mean spectra of the points ${P_{i}, i = 1, N_{p}}$ can be expressed as

Figure 13. EQUATION_DISPLAY

{\bar{g}}_{s} (ω) = \frac{1}{N_{p}} \sum_{i = 1, N_{p}}^{} g (P_{i}; ω)

(534)

The temporal-spatial mean is defined as

Figure 14. EQUATION_DISPLAY

{\bar{g}}_{t s} (ω) = \frac{1}{N_{p}} \sum_{i = 1, N_{p}}^{} {\bar{g}}_{t} (P_{i}; ω)

(535)