An improved sample-and-hold reconstruction procedure for estimation of power spectra from LDA data

Abstract

Techniques for deriving the auto or power spectrum (PSD) of turbulence from laser Doppler anemometry (LDA) measurements are reviewed briefly. The low pass filter and step noise errors associated with the sample-and-hold process are considered and a discrete version of the low pass filter for the resampled signal is derived. This is then used to develop a procedure by which the PSD estimates obtained from sample and hold measurements can be corrected. The application of the procedures is examined using simulated data and the results show that the frequency range of the analysis can be extended beyond the Nyquist frequency based on the mean sample rate. The results are shown to be comparable to those obtained using the method of Nobach et al. (1998) but the new procedures are more straightforward to implement. The technique is then used to determine the PSD of real LDA data and the results are compared with those from a hot wire anemometer.

Appendix

Discrete low pass filter

Let ${\text{R}}_{{{\text{rr}}}} {\text{[n]}} \equiv {\text{R}}_{{{\text{rr}}}} {\left( {{\text{nT}}_{{\text{e}}} } \right)}$ (where T_e is the resampling period) be the discrete-time correlation of r(t) obtained by sampling the continuous-time correlation R_rr (τ)

$${\text{R}}_{{{\text{rr}}}} [{\text{n}}] = {\text{ e}}^{{{\text{ - }}\lambda {\left| {\text{n}} \right|}{\text{T}}_{{\text{e}}} }} {\text{ }}{\left( {{\text{R}}_{{{\text{uu}}}} (0) - \frac{\lambda } {2}{\int {{\text{e}}^{{ - \lambda {\left| \theta \right|}}} } }{\text{R}}_{{{\text{uu}}}} (\theta ){\text{d}}\theta } \right)} + \frac{\lambda } {2}{\int {{\text{e}}^{{ - \lambda {\left| \theta \right|}}} {\text{R}}_{{{\text{uu}}}} ({\left| {\text{n}} \right|}{\text{T}}_{{\text{e}}} - \theta ){\text{d}}\theta } }$$

(A1)

Then, the continuous-frequency PSD ${\text{G}}_{{{\text{rr}}}} {\left( {{\text{e}}^{{{\text{j2}}\pi {\text{fT}}_{{\text{e}}} }} } \right)}$ of the discrete-time correlation R_rr [n]

$${\text{G}}_{{{\text{rr}}}} {\left( {{\text{e}}^{{{\text{j2}}\pi {\text{fT}}_{{\text{e}}} }} } \right)} = {\sum\limits_{\text{n}} {{\text{R}}_{{{\text{rr}}}} [{\text{n}}]} }{\text{e}}^{{{\text{ - j2}}\pi {\text{fnT}}_{{\text{e}}} }} $$

(A2)

may be written according to (2) as

$$\begin{aligned} & {\text{G}}_{{{\text{rr}}}} {\left( {{\text{e}}^{{{\text{j2}}\pi {\text{fT}}_{{\text{e}}} }} } \right)} \quad = {\text{f}}_{{\text{e}}} {\sum\limits_{\text{k}} {{\text{G}}_{{{\text{rr}}}} {\left( {{\text{f - kf}}_{{\text{e}}} } \right)}} } \\ & \quad = {\left| {{\text{L}}_{{\text{c}}} {\left( {{\text{e}}^{{{\text{j2}}\pi {\text{fT}}_{{\text{e}}} }} } \right)}} \right|}^{2} {\left( {{\text{G}}_{{{\text{uu}}}} {\left( {{\text{e}}^{{{\text{j2}}\pi {\text{fT}}_{{\text{e}}} }} } \right)} + {\text{G}}_{{{\text{ss}}}} {\left( {{\text{e}}^{{{\text{j2}}\pi {\text{fT}}_{{\text{e}}} }} } \right)}} \right)} \\ \end{aligned} $$

(A3)

where ${\text{f}}_{{\text{e}}} {\text{ = }}\frac{{\text{1}}} {{{\text{T}}_{{\text{e}}} }},$ where ${\text{G}}_{{{\text{uu}}}} {\left( {{\text{e}}^{{{\text{j2}}\pi {\text{fT}}_{{\text{e}}} }} } \right)}$ is the Fourier transform of the discrete-time auto-correlation function R_uu [n] and where ${\text{G}}_{{{\text{ss}}}} {\left( {{\text{e}}^{{{\text{j2}}\pi {\text{fT}}_{{\text{e}}} }} } \right)}$ is the Fourier transform of the discrete-time auto-correlation function of the step noise R_ss [n]. It is important to note that ${\text{L}}_{{\text{c}}} {\left( {{\text{e}}^{{{\text{j2}}\pi {\text{fT}}_{{\text{e}}} }} } \right)}$ defined in Eq. (A3) may be approximated according to

$${\left| {{\text{L}}_{{\text{c}}} {\left( {{\text{e}}^{{{\text{j2}}\pi {\text{fT}}_{{\text{e}}} }} } \right)}} \right|}^{2} \cong {\sum\limits_{\text{k}} {{\left| {{\text{L}}_{{\text{c}}} {\left( {{\text{f}} - {\text{kf}}_{{\text{e}}} } \right)}} \right|}^{2} } }$$

(A4)

if the maximum frequency f_max we wish to estimate in the PSD ${\text{G}}_{{{\text{uu}}}} {\left( {{\text{e}}^{{{\text{j2}}\pi {\text{fT}}_{{\text{e}}} }} } \right)}$ is such that

$${\left| {{\text{L}}_{{\text{c}}} {\left( {{\text{f}}_{{{\text{max}}}} - {\text{f}}_{{\text{e}}} } \right)}} \right|}^{2} < < {\left| {{\text{L}}_{{\text{c}}} {\left( {{\text{f}}_{{{\text{max}}}} } \right)}} \right|}^{2} $$

(A5)

or equivalently from Eq. (5)

$${\text{f}}_{{\text{e}}} > > {\text{2f}}_{{{\text{max}}}} $$

(A6)

If this assumption is established, we note ${\text{ \ifmmode\expandafter\tilde\else\expandafter\~\fi{L}}}_{{\text{c}}} {\text{(f)}} = {\left| {{\text{L}}_{{\text{c}}} {\text{(f)}}} \right|}^{2} \;{\text{and}}\;{\text{ \ifmmode\expandafter\tilde\else\expandafter\~\fi{l}}}_{{\text{c}}} (\tau )$ the inverse Fourier transform of ${\text{ \ifmmode\expandafter\tilde\else\expandafter\~\fi{L}}}_{{\text{c}}} {\text{(f)}}{\text{.}}$ It is easy to show from Eq. (5) that

$${\text{ \ifmmode\expandafter\tilde\else\expandafter\~\fi{l}}}_{{\text{c}}} (\tau ) = \frac{\lambda } {2}{\text{e}}^{{ - \lambda {\left| \tau \right|}}} $$

(A7)

Consequently, the Fourier transform of the discrete-time impulse response ${\text{ \ifmmode\expandafter\tilde\else\expandafter\~\fi{l}}}_{{\text{c}}} {\text{[n]}} \equiv {\text{ \ifmmode\expandafter\tilde\else\expandafter\~\fi{l}}}_{{\text{c}}} {\left( {{\text{nT}}_{{\text{e}}} } \right)}$ defined by

$${\text{ \ifmmode\expandafter\tilde\else\expandafter\~\fi{L}}}_{{\text{c}}} {\left( {{\text{e}}^{{{\text{j2}}\pi {\text{fT}}_{{\text{e}}} }} } \right)} = {\left| {{\text{L}}_{{\text{c}}} {\left( {{\text{e}}^{{{\text{j2}}\pi {\text{fT}}_{{\text{e}}} }} } \right)}} \right|}^{2} = {\text{T}}_{{\text{e}}} {\sum\limits_{\text{n}} {{\text{ \ifmmode\expandafter\tilde\else\expandafter\~\fi{l}}}_{{\text{c}}} [n]} }{\text{e}}^{{{\text{ - j2}}\pi {\text{fnT}}_{{\text{e}}} }} $$

(A8)

may be written

$${\left| {{\text{L}}_{{\text{c}}} {\left( {{\text{e}}^{{{\text{j2}}\pi {\text{fT}}_{{\text{e}}} }} } \right)}} \right|}^{2} \cong \frac{{{\text{T}}_{{\text{e}}} \lambda }} {2} \cdot \frac{{1 - {\text{e}}^{{ - 2\lambda {\text{T}}_{{\text{e}}} }} }} {{1 - 2\cos {\left( {2\pi {\text{fT}}_{{\text{e}}} } \right)}{\text{e}}^{{ - \lambda {\text{T}}_{{\text{e}}} }} + {\text{e}}^{{ - 2\lambda {\text{T}}_{{\text{e}}} }} }}$$

(A9)

If f_max is chosen for instance higher than λ, the numerator of Eq. (A8) may thus be approximated as

$${\left| {{\text{L}}_{{\text{c}}} {\left( {{\text{e}}^{{{\text{j2}}\pi {\text{fT}}_{{\text{e}}} }} } \right)}} \right|}^{2} \cong \frac{{{\left( {{\text{T}}_{{\text{e}}} \lambda } \right)}^{2} }} {{1 - 2\cos {\left( {2\pi {\text{fT}}_{{\text{e}}} } \right)}{\text{e}}^{{ - \lambda {\text{T}}_{{\text{e}}} }} + {\text{e}}^{{ - 2\lambda {\text{T}}_{{\text{e}}} }} }}$$

(A10)

Lastly, using Eqs. (A1), (A2), (A3) and (A9) leads to

$${\text{G}}_{{{\text{ss}}}} {\left( {{\text{e}}^{{{\text{j}}2\pi {\text{fT}}_{{\text{e}}} }} } \right)} = {\text{ }}\frac{{\text{2}}} {{\lambda {\text{T}}_{{\text{e}}} }}{\text{ }}{\left( {{\text{R}}_{{{\text{uu}}}} (0) - \frac{\lambda } {2}{\int {{\text{e}}^{{ - \lambda {\left| \theta \right|}}} } }{\text{R}}_{{{\text{uu}}}} (\theta ){\text{d}}\theta } \right)}\,.$$

(A11)