11.1 Limitations of measured data

The very act of measurement places fundamental limitations on the information we can extract. Whatever recording system we use will have a limited range and resolution, and therefore a limited dynamic range. In a digital measurement system, the dynamic range is stated in bits. For example, a 12-bit system is one that can sense 2¹² (4096) distinct signal levels, implying a dynamic range of 20log₁₀ 2¹²=72 dB. Common also are 16- and 24-bit systems which imply dynamic ranges of 96 and 144 dB, respectively. Obviously the bit-resolution places a fundamental limit on the accuracy of a measurement, and the relative accuracy is optimized if a signal is amplified or attenuated so that its fluctuations span, as nearly as possible, the full range of a measurement system.

A further limitation is that the signal will have to be sampled at a finite rate f_s=1/Δt, where Δt is the time between successive samples and is referred to as the sampling period. To define a sinusoidal waveform requires at least two samples per period (e.g., one at each successive peak and trough). Thus the highest frequency that can be inferred from a sampled signal is limited to half the sampling rate. This is referred to as the Nyquist frequency or Nyquist criterion, after the Swedish-American electronics engineer Harry Nyquist [3]. Finally, we can only measure the signal for a finite period of time T_o. As such the measurement cannot be used to infer the presence of any frequency with a period greater than the sampling time, at least not without making assumptions about the behavior of the signal before we started or after we finished measuring. Thus the lowest frequency we can unambiguously infer from our measurement is 2π/T_orad/s, regardless of whether lower frequencies are present.

It is particularly important to be aware of the consequences of not meeting the Nyquist criterion if one is interested in extracting spectral or time information about a signal. Consider, for example, the broadband turbulent velocity signal shown in Fig. 11.1A measured in a boundary layer much like that in our rotor example. We know from our analysis of the spectrum of this same signal in Chapter 8 (see Fig. 8.3C) that it contains little, if any, power at frequencies above 20 kHz. Sampling at 50 kHz, as shown by the black trace in Fig. 11.1A, therefore meets the Nyquist criterion, and the sampled signal contains a complete representation of the highest frequency fluctuations. Sampling at the much lower frequency of 2 kHz, shown with the gray trace, not only misrepresents the content of the signal at frequencies greater than the Nyquist but also corrupts the information we can obtain at lower frequencies.

Exactly how this occurs becomes clearer if we consider a simple sine wave. The 1900 Hz wave of Fig. 11.1B is well defined when sampled at 50 kHz. Sampling at 2 kHz, however, produces data points that combine to form a spurious and much lower frequency waveform of 100 Hz even though each sample still accurately represents the level of the signal at the instant it was taken. This misidentification of frequency content, called aliasing, is a serious problem. With only the 2 kHz sampled data we will have no way of knowing whether the 100 Hz signal was actually present or not. Furthermore, if we are measuring a broadband signal, like the turbulent velocity in Fig. 11.1A, the contribution at 100 Hz that comes from aliasing may mask any actual content we have at that frequency.

If all we are interested in is simple statistical information (such as the mean-square) which does not involve the signal sequence, then aliasing is of no concern. All that matters is that we take sufficient independent samples to accurately estimate the expected value. We can see this in Fig. 11.1B where the amplitude, and therefore the mean square, of the two sampled signals is the same. If we are interested in frequency content or time-delay information, then avoiding aliasing is critical. Even if one can estimate the maximum frequency of a physical quantity to be measured, the aliasing of high-frequency interference unintentionally present in the sensor signal may doom a measurement or produce misleading results. For this reason, most digital measurement systems include analog low-pass filters that are applied to a signal before it is measured to ensure that the Nyquist criterion is satisfied.

In the above discussion we have made reference to time signals and frequency analysis, but the same limitations apply when the variation is in a signal that exists in space and we want to know its wavenumber content. For example, Fig. 11.2 shows an instantaneous velocity vector field measured in the bottom of the boundary layer parallel to the wall under the tips of the rotor blade in Fig. 10.41. Consider the fluctuating velocities seen along x₃ at x₁=0 as shown in Fig. 11.2B. These are measured every 2.23 mm over a total length of 240 mm. The Nyquist criterion therefore restricts us to inferring information at wavenumbers less than 2π/0.00223=2818 rad/m, and smaller spatial scales will cause aliasing. At the same time we can only infer information at wavenumbers down to 2π/0.24=26 rad/m.

11.2 Uncertainty

All experimental measurements are subject to some degree of error. The errors may be random, in that they contribute to the scatter of a measurement about the actual value, or bias where the average of a sequence of measurements of the same quantity does not converge to its actual value. In order for an experiment to be useful we need to have a feel for the typical size of the errors in its results. Uncertainty analysis is the term given to the methods used to estimate errors and then track their propagation through the measurement and data analysis process to a final derived result. We denote the error in terms of an uncertainty interval or band. The interval is chosen so that there is a specified probability of the true value lying within it. The specified probability is generally taken to be 95%, unless other considerations (such as safety) dictate a more stringent standard.

An uncertainty interval is commonly represented using the ± sign as in “the Mach number was measured to be 0.06±0.001” meaning that there is a 95% chance that the true Mach number was between 0.059 and 0.061. Symbolically, uncertainties are indicated using the δ[] notation and may be stated in absolute (e.g., δ[M]=0.001) or relative terms (e.g., δ[M]/M=0.017, or 1.7%). Note that we use square brackets here to distinguish the notation used for uncertainty from that used for the Dirac delta function, δ( ).

In general, any experimental result R_o will be a function of some number of raw measurements m_n, i.e., measurements not derived from any other. Each measurement will be subject to an unknown error ɛ_n. If we assume the errors are small, then the error in the result ɛ_R will be given by

${\varepsilon }_R\cong {\displaystyle \sum _{n=1}{\varepsilon }_n\frac{\partial R_o}{\partial m_n}}$

where we are ignoring second- and higher-order terms in the Taylor expansion of R_o. In uncertainty analysis we are interested in estimating the typical error, which can be obtained by taking the mean square of Eq. (11.2.1) to give

$\overline{{\varepsilon }_R^2}\cong {{\displaystyle \sum _{n=1}\overline{{\varepsilon }_n^2}\left(\frac{\partial R_o}{\partial m_n}\right)}}^2$

where we have assumed that the errors in different measurements are uncorrelated. If the errors are all normally distributed then the uncertainty at 95% probability is proportional to the root mean square error, where the constant of proportionality of 1.96 is usually rounded to 2. So, we are 95% certain that the actual value of R_o lies between −δ[R_o] and δ[R_o] of the measured value, where

$\delta \left[R_o\right]\cong \sqrt{{\displaystyle \sum _{n=1}{\left[\delta \left[m_n\right]\frac{\partial R_o}{\partial m_n}\right]}^2}}$

Eq. (11.2.3) is a broadly useful tool for tracking the propagation of uncertainties through the analysis process. However, it is important to keep in mind that it rests on assumptions that may not be valid in some situations. Indeed, we will meet such a situation in the next section.

As an example of the application of Eq. (11.2.3), consider estimating an uncertainty interval for the measurement of free stream velocity in the rotor experiment of Fig. 10.41. The raw measurements contributing to this result are the dynamic pressure of the free stream $p_{\mathit{dyn}}=\mathrm{½}{\rho }_o{U}_{\infty }^2$ measured using a Pitot static tube and its absolute temperature and pressure, T_e and p_o, used to determine the density (ρ_o=p_o/RT_e). We have that

$U_{\infty }=\sqrt{\frac{2R{T}_e{p}_{\mathit{dyn}}}{p_o}}$

and thus, since the errors in these measurements will be independent,

$\delta \left[U_{\infty }\right]=\sqrt{{\left[\delta \left[p_{\mathit{dyn}}\right]\left(\frac{1}{2}\frac{U_{\infty }}{p_{\mathit{dyn}}}\right)\right]}^2+{\left[\delta \left[T_e\right]\left(\frac{1}{2}\frac{U_{\infty }}{T_e}\right)\right]}^2+{\left[\delta \left[p_o\right]\left(-\frac{1}{2}\frac{U_{\infty }}{p_o}\right)\right]}^2}$

In this particular experiment typical values were U_∞=20 m/s, T_e=290 K, p_o=945 mBar, and p_dyn=227 Pa. The raw measurement uncertainties δ[p_dyn], δ[T_e], and δ[p_o] are obtained from the specifications and/or calibrations of the transducers used, the acquisition hardware, and the judgment of the experimentalist. Reasonable values in this case might be 2.5 Pa, 0.2 K, and 0.5 mBar, respectively, giving,

$\delta \left[U_{\infty }\right]=\sqrt{0.0121+0.00005+0.00003}=0.11\mathrm{m}/\mathrm{s}$

where the terms under the square root are written in the same order as in Eq. (11.2.5). We see that the uncertainty is dominated by the dynamic pressure measurement, and thus, if we wanted to improve our accuracy then we would invest time or money here. Note that we did not consider uncertainty in the gas constant R. This would only be necessary if we were expecting to achieve a relative uncertainty comparable to that with which R is known.

Several general observations can be made here. First, it is sometimes easier and more reliable to calculate the derivatives needed for the uncertainty calculation numerically. In particular, with measurements processed using a computer program this can be done by perturbing each of the raw measurements input to the program, in turn, by its uncertainty. The change in the result produced by each trial will approximate δ[m_n]∂R_o/∂m_n. Second, informed guesswork by the experimentalist is often necessary to estimate the uncertainty in a raw measurement (and, if necessary, the correlation between raw measurements). This is fine. The experimentalist still has a better idea of this error than anyone else and, if the measurement is worth making, the uncertainty will be a small proportion and so possible inaccuracy within the uncertainty estimate itself is not the main concern. Finally, the root mean square addition of uncertainties ensures that we will have a tendency to underestimate the uncertainty in a result, since any neglected contribution would only have increased its value. As such, it is the experimentalist's responsibility to substitute a credible value for the uncertainty if they feel that a computed estimate is unreasonably small (though any such estimate should of course be recorded).

11.3 Averaging and convergence

Averaging is a common feature of aeroacoustic measurement since many of the properties in which we are interested are defined as expected values. This means we need to know how many averages are necessary to determine the desired result to within a certain accuracy, i.e., the uncertainty band. Consider, as an example, the uncertainty of a mean value that is being measured from a series of N samples of a quantity a:

$〈\overline{a}〉=\frac{1}{N}{\displaystyle \sum _{n=1}^N}{a}_n$

The notation $〈\overline{a}〉$is used to recognize that, being determined from a limited number of samples, this is only an estimate of the true mean value $\overline{a}$. Since Eq. (11.3.1) describes the functional relationship between the mean estimator and the samples from which it is calculated, it appears quite possible to determine the uncertainty in $〈\overline{a}〉$ from the range of fluctuations in a_n using the error propagation equation (11.2.3). However, for almost all statistical quantities the nonlinear and/or correlation terms ignored in this equation are significant, and the analysis becomes unwieldy if these are included. We therefore use a different approach, directly deriving an equation for the variance in the error. For example, for the mean value we determine the variance of its estimated value as

$\begin{array}{ll}\overline{{\varepsilon }^2(〈\overline{a}〉})& =E\left[{\left(〈\overline{a}〉-\overline{a}\right)}^2\right]=E\left[{\left(\left\{\frac{1}{N}{\displaystyle \sum _{n=1}^N}{a}_n\right\}-\overline{a}\right)}^2\right]=E\left[{\left(\frac{1}{N}{\displaystyle \sum _{n=1}^N}\left(a_n-\overline{a}\right)\right)}^2\right]\hfill \\ & =E\left[{\left(\frac{1}{N}{\displaystyle \sum _{n=1}^N}{a}_n^{\prime }\right)}^2\right]\hfill \end{array}$

where the prime denotes the fluctuating part. Replacing the square with a double sum we can write this as

$\overline{{\varepsilon }^2\left(〈\overline{a}〉\right)}=E\left[\frac{1}{N^2}{\displaystyle \sum _{n=1}^N}{\displaystyle \sum _{p=1}^N}{a}_n^{\prime }{a}_p^{\prime }\right]=\frac{1}{N^2}{\displaystyle \sum _{n=1}^N}{\displaystyle \sum _{p=1}^N}E\left[a_n^{\prime }{a}_p^{\prime }\right]$

At this point we can make one of two choices. The first is to assume that the samples are all independent of each other, such as would be the case if they were collected by occasionally checking the value of some reference quantity, in which case $E\left[a_n^{\prime }{a}_p^{\prime }\right]=\overline{{a^{\prime }}^2}{\delta }_{np}$ (where δ_np is the Kronecker delta) and Eq. (11.3.3) becomes

$\overline{{\varepsilon }^2(〈\overline{a}〉})=\frac{\overline{{a^{\prime }}^2}}{N}$

Assuming the error in the mean is normally distributed, the uncertainty interval at 95% odds is determined as twice its standard deviation.

$\delta \left[〈\overline{a}〉\right]=\frac{2\sqrt{\overline{{a^{\prime }}^2}}}{\sqrt{N}}$

So, uncertainty decreases as the inverse square root of the number of independent samples used in forming the average.

The second option in evaluating Eq. (11.3.3) is to assume that the samples are taken in rapid succession at equal intervals Δt as they would be if they were part of a time-resolved measurement. In this case $E\left[a_n^{\prime }{a}_p^{\prime }\right]=\overline{{a^{\prime }}^2}{\rho }_{\mathit{aa}}\left(\left[n-p\right]\mathrm{\Delta }t\right)$, where ρ_aa is the correlation coefficient function introduced in Eq. (8.4.5), and so we have

$\overline{{\varepsilon }^2\left(〈\overline{a}〉\right)}=E\left[\frac{1}{N^2}{\displaystyle \sum _{n=1}^N}{\displaystyle \sum _{p=1}^N}{a}_n^{\prime }{a}_p^{\prime }\right]=\frac{1}{N^2}{\displaystyle \sum _{n=1}^N}{\displaystyle \sum _{p=1}^N}\overline{{a^{\prime }}^2}{\rho }_{\mathit{aa}}\left(\left[n-p\right]\mathrm{\Delta }t\right)$

Introducing the index q=n−p we can reorganize the summations as

$\overline{{\varepsilon }^2\left(〈\overline{a}〉\right)}=\frac{\overline{{a^{\prime }}^2}}{N^2\mathrm{\Delta }t}{\displaystyle \sum _{q=1-N}^{N-1}}\left(N-\left|q\right|\right){\rho }_{\mathit{aa}}\left(q\mathrm{\Delta }t\right)\mathrm{\Delta }t$

Now, from Eq. (8.4.5), ${\displaystyle \sum _{q=1-N}^{N-1}}\rho \left(q\mathrm{\Delta }t\right)\mathrm{\Delta }t$ is equal to twice the integral timescale $\mathcal{T}$, assuming $\mathrm{\Delta }t\ll \mathcal{T}$ and the total sampling time T_o=$N\mathrm{\Delta }t\gg \mathcal{T}$. Applying these assumptions to Eq. (11.3.7) gives,

${\displaystyle \sum _{q=1-N}^{N-1}}\left(N-\left|q\right|\right){\rho }_{\mathit{aa}}\left(q\mathrm{\Delta }t\right)\mathrm{\Delta }t\cong {\displaystyle \sum _{q=1-N}^{N-1}}N{\rho }_{\mathit{aa}}\left(q\mathrm{\Delta }t\right)\mathrm{\Delta }t=2N\mathcal{T}$

and so we have

$\overline{{\varepsilon }^2\left(〈\overline{a}〉\right)}\cong \frac{\overline{{a^{\prime }}^2}}{T_o}2\mathcal{T}$

and an uncertainty interval of

$\delta \left[〈\overline{a}〉\right]=\frac{2\sqrt{\overline{{a^{\prime }}^2}}}{\sqrt{T_o/2\mathcal{T}}}$

In this case the uncertainty is independent of the number of samples we take if we are measuring for a fixed time. This can be visualized in terms of the rapidly sampled signal (black symbols) of Fig. 11.1A. Averaging the 100 samples of this signal between t=0.004 and 0.006 gives a poor estimate of $\overline{u_1}$. However, since the data taken already fully define the signal, we cannot improve this estimate by sampling faster to take more data over this period. Comparing Eqs. (11.3.10), (11.3.5) shows that for time-resolved data the effective number of independent samples is given by the total sampling time divided by twice the integral time scale. A safe rule in general is to take the effective number of independent samples as the minimum of N and $T_o/2\mathcal{T}$.

In a similar fashion we can derive uncertainty relations for other statistics including the mean-square and the cross correlation [4]. These are summarized in Table 11.1. Note that these formulae assume independent samples, and that the cross correlation and coefficient expressions can equally well be applied to space or time-delay correlation function estimates of single variables or pairs of variables simply by appropriately assigning a and b. Technically, the effective number of independent samples for the second-order statistics depends on the integral scale of the square or product of correlation functions. As a practical matter, however, this distinction is often ignored, and the minimum of N and $T_o/2\mathcal{T}$ is still used.

Table 11.1

Averaging uncertainties

Quantity	Estimator	Averaging uncertainty
Mean	$〈\overline{a}〉=\frac{1}{N}\sum _{n=1}^N{a}_n$	$\delta \left[〈\overline{a}〉\right]=\frac{2\sqrt{\overline{{a^{\prime }}^2}}}{\sqrt{N}}$
Mean square fluctuation	$〈\overline{{a^{\prime }}^2}〉=\frac{1}{N}\sum _{n=1}^N{a^{\prime }}_n^2$	$\delta \left[〈\overline{{a^{\prime }}^2}〉\right]=\frac{2\sqrt{2}\overline{{a^{\prime }}^2}}{\sqrt{N}}$
RMS	$\sqrt{〈\overline{{a^{\prime }}^2}〉}=\sqrt{\frac{1}{N}\sum _{n=1}^{N}a{\prime }_n^2}$	$\delta \left[\sqrt{〈\overline{{a^{\prime }}^2}〉}\right]=\frac{\sqrt{2}\sqrt{\overline{{a^{\prime }}^2}}}{\sqrt{N}}$
Cross correlation	$〈\overline{a\prime b\prime }〉=\frac{1}{N}\sum _{n=1}^N{a}_n^{\prime }{b}_n^{\prime }$	$\delta \left[〈\overline{a\prime b\prime }〉\right]=\frac{2\sqrt{\overline{{a^{\prime }}^2}}\sqrt{\overline{{b^{\prime }}^2}}}{\sqrt{N}}\sqrt{\left(1+{\rho }_{\mathit{ab}}^2\right)}$
Cross correlation coefficient	$〈{\rho }_{\mathit{ab}}〉=\frac{〈\overline{a\prime b\prime }〉}{\sqrt{〈\overline{{a^{\prime }}^2}〉〈\overline{b^2}〉}}$	$\delta \left[〈{\rho }_{\mathit{ab}}〉\right]=\frac{2}{\sqrt{N}}\left(1-{\rho }_{\mathit{ab}}^2\right)$

11.4 Numerically estimating fourier transforms

In Chapter 8 we introduced the autospectrum and the cross spectrum as important measures of turbulent flows and the acoustic fields that they generate. The measurement of these functions, and the quantities that can be inferred from them, is therefore often the central focus of aeroacoustics experiments. In this and the following sections we develop the tools needed to best estimate these functions and related quantities using sampled data. As a first step we introduce in this section the discrete Fourier transform (DFT) and its inverse discrete Fourier transform (IDFT), as approximations to the continuous Fourier transform we use in analysis. The DFT is defined as

$A_m\text{≡}{\displaystyle \sum _{n=1}^N}{a}_n{e}^{-\frac{2\pi i\left(n-1\right)\left(m-1\right)}{N}}\text{≡}\mathrm{D}\mathrm{F}\mathrm{T}\left(a_n,m\right)$

for $m=1$ to N. The IDFT is defined as

$a_n\text{≡}\frac{1}{N}{\displaystyle \sum _{m=1}^N}{A}_m{e}^{\frac{2\pi i\left(n-1\right)\left(m-1\right)}{N}}\text{≡}\mathrm{IDFT}\left(A_m,n\right)$

for $n=1$ to N. Note that these specific definitions are the same as those used by the Matlab programming environment¹ and other popular software tools, and that the IDFT is the inverse of the DFT so that

$a_n=\mathrm{IDFT}\left(\mathrm{D}\mathrm{F}\mathrm{T}\left(a_n,m\right),n\right)$

Eqs. (11.4.1), (11.4.2) are rarely computed explicitly as written since this is an expensive calculation requiring O(N²) operations. Instead the Fast Fourier Transform algorithm [5] is used which takes advantage of efficiencies which become possible when N is a composite number, and particularly when it is a power of 2. This reduces the computational effort to O(N log N)—a huge saving when large data sets are involved.

Consider now the definition of the continuous Fourier transform in time, Eq. (8.4.1), applied to a signal that lasts a finite time of T_o seconds.

${\tilde{a}}^{\left(s\right)}\left(\omega \right)=\frac{1}{2\pi }\underset{0}{\overset{T_o}{\int }}a\left(t\right)e^{i\omega t}dt$

We use the symbol ${\tilde{a}}^{\left(s\right)}\left(\omega \right)$ since this is slightly different than Eq. (8.4.1) where the time range of the integral extends from −T to T. Suppose that the signal is formed from N samples of the quantity a taken at the beginning of a series of regular time intervals Δt, i.e., a_n=a([n−1]Δt) for n=1 to N so that T_o=NΔt. To apply the definition to this signal we replace the integral by a summation

$〈\tilde{a}\left(\omega \right)〉=\frac{1}{2\pi }{\displaystyle \sum _{n=1}^N}{a}_n{e}^{i\omega \left(n-1\right)\mathrm{\Delta }t}\mathrm{\Delta }t$

where $〈\tilde{a},\left(\mathit{\omega }\right)〉$ indicates that we are using this discretization of Eq. (11.4.3) to provide an estimate of the continuous Fourier transform $\tilde{a}\left(\mathit{\omega }\right)$ as defined in Eq. (8.4.1), the function we are really interested in. We will address exactly how $〈\tilde{a},\left(\mathit{\omega }\right)〉$ is different than $\tilde{a}\left(\mathit{\omega }\right)$ in Section 11.5. To apply Eq. (11.4.4) numerically we also need to discretize the result of the Fourier transform as $〈{\tilde{a}}_m〉=〈\tilde{a}\left(\left(m-1\right)\mathrm{\Delta }\omega \right)〉$ for m=1 to N. Note that frequency resolution Δω is also the smallest nonzero frequency at which we are going to get a result, so we choose this to be equal to the lowest frequency that can be unambiguously determined from the sampled time signal, 2π/T_o. Thus,

$〈{\tilde{a}}_m〉=\frac{1}{2\pi }{\displaystyle \sum _{n=1}^N}{a}_n{e}^{i\left(m-1\right)\mathrm{\Delta }\omega \left(n-1\right)\mathrm{\Delta }t}\mathrm{\Delta }t$

Now ΔωΔt=2πΔt/T_o=2π/N, and so,

$〈{\tilde{a}}_m〉=\frac{1}{2\pi }{\displaystyle \sum _{n=1}^N}{a}_n{e}^{\frac{2\pi i\left(n-1\right)\left(m-1\right)}{N}}\mathrm{\Delta }t$

Comparing this with Eq. (11.4.1), we see that the numerical estimate of the Fourier transform can be calculated as

$〈{\tilde{a}}_m〉=\frac{\mathrm{\Delta }t}{2\pi }\mathrm{D}\mathrm{F}{\mathrm{T}}^{⁎}\left(a_n,m\right)$

where the asterisk denotes the complex conjugate of the DFT. If we restrict ourselves to times t from 0 to T_o then the inverse of the continuous Fourier transform of Eq. (11.4.3) is given by,

$a\left(t\right)={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}{\tilde{a}}^{\left(s\right)}\left(\omega \right)e^{-i\omega t}d\omega$

and the equivalent inverse numerical transform to Eq. (11.4.7) is

$a_n=\frac{2\pi }{N\mathrm{\Delta }t}\sum _{m=1}^N〈{\tilde{a}}_m〉e^{-\frac{2\pi i\left(m-1\right)\left(n-1\right)}{N}}=\frac{2\pi }{\mathrm{\Delta }t}{\mathrm{IDFT}}^{⁎}\left(〈{\tilde{a}}_m〉,n\right)$

where Δt=2π/NΔω.

Consider the example shown in Fig. 11.3 of a time domain signal, represented by samples a_n, and the values of the transform $〈{\tilde{a}}_m〉$ it implies. The signal is a simple square pulse defined by 32 samples. The sampling period Δt is 1 s so that the frequency spacing in the Fourier domain is 2π/32 rad/s. Both a_n and $〈{\tilde{a}}_m〉$ are plotted against their indices.

Before discussing its individual elements, it is important to note that transform defined by Eq. (11.4.6) is actually periodic. Since adding or subtracting any integer multiple of 2π to the exponent does not change its value, then $〈{\tilde{a}}_m〉=〈{\tilde{a}}_{m+N}〉=〈{\tilde{a}}_{m-N}〉=〈{\tilde{a}}_{m+2N}〉$, and so on. The frequency values for which we had derived this expression, and that are plotted in Fig. 11.3B, thus represent a single period of an infinitely repeating function. Unexpectedly, the same is also true for the inverse transform of Eq. (11.4.9) which tells us that a_n=a_n+pN, where p is any integer. In other words, we have inadvertently assumed in our formulation of the numerical Fourier transform that the samples we measure form precisely one period of a periodic time variation. Since this is not usually the case, it can be a source of significant bias error, termed broadening. As will be discussed later, the source of this assumption was the discretization of the transform into frequency intervals.

Consider now the elements of $〈{\tilde{a}}_m〉$ for m=1 to N shown in Fig. 11.3B. The first element $〈{\tilde{a}}_1〉$ is simply 1/2π times the mean value of the sampled signal since, from Eq. (11.4.6),

$〈{\tilde{a}}_1〉=\frac{1}{2\pi }{\displaystyle \sum _{n=1}^N}{a}_n\mathrm{\Delta }t$

This element is therefore always real. Another element with no imaginary part is that corresponding to the Nyquist frequency (π/Δt in terms of angular frequency) which occurs at index m=N/2+1 (equal to 17 in the example). Here Eq. (11.4.6) reduces to

$〈{\tilde{a}}_{N/2+1}〉=\frac{1}{2\pi }{\displaystyle \sum _{n=1}^N}{a}_n{e}^{\pi i\left(n-1\right)}\mathrm{\Delta }t=\frac{1}{2\pi }{\displaystyle \sum _{n=1}^N}{a}_n{\left(-1\right)}^{n-1}\mathrm{\Delta }t$

Since this is always real we can conclude that the numerical Fourier transform extracts no useful phase information at the Nyquist frequency.

Fig. 11.3B shows that the remainder of the transform has a conjugate symmetry with $〈{\tilde{a}}_2〉={〈{\tilde{a}}_N〉}^{⁎},〈{\tilde{a}}_3〉={〈{\tilde{a}}_{N-1}〉}^{⁎},\dots 〈{\tilde{a}}_m〉={〈{\tilde{a}}_{N-m+2}〉}^{⁎}$. This is a consequence of the fact that the signal a_n is real since, from Eq. (11.4.6),

$〈{\tilde{a}}_{N-m+2}〉=\frac{1}{2\pi }{\displaystyle \sum _{n=1}^N{a}_n{e}^{\frac{2\pi i\left(n-1\right)\left(N-m+2-1\right)}{N}}}\mathrm{\Delta }t=\frac{1}{2\pi }{\displaystyle \sum _{n=1}^N{a}_n{e}^{\frac{2\pi i\left(n-1\right)\left(-m+1\right)}{N}}}\mathrm{\Delta }t=\frac{1}{2\pi }{\displaystyle \sum _{n=1}^N{a}_n{e}^{-\frac{2\pi i\left(n-1\right)\left(m-1\right)}{N}}}\mathrm{\Delta }t$

which is the conjugate of $〈{\tilde{a}}_m〉$ for real a_n. The axis of symmetry axis occurs when m=N−m+2, corresponding to m=N/2+1, the Nyquist frequency. Since the numerical transform is periodic we can, and usually do, ascribe the reflected spectral values from m=N/2+2 to N to negative frequencies corresponding to m=−N/2+2 to 0, as illustrated in Fig. 11.4C. In this way the numerical transform is put in a form that is explicitly seen to match the conjugate symmetry of the continuous Fourier transform about zero frequency.

The above discussion demonstrates that the numerical Fourier transform contains no new information at frequencies greater than the Nyquist, as we would expect given our discussion in Section 11.1. Furthermore we see that $〈{\tilde{a}}_m〉$ is defined by N unique values (2 real numbers and N/2−1 complex numbers requiring 2 values each) matching the information content of the original sampled signal a_n. Table 11.2 summarizes the structure of the Fourier transform vector $〈{\tilde{a}}_m〉$. Note that any vector that does not match this structure will imply that the time signal samples a_n are complex numbers.

Table 11.2

Structure of the numerical Fourier transform $〈{\tilde{a}}_m〉$

Index	Description
m=1	Mean value (real)
$2\le m\le \frac{N}{2}$	Numerical Fourier transform values for frequencies $\omega =\left(m-1\right)\frac{2\pi }{T_o}$
$\frac{N}{2}+1$	Nyquist frequency value (real)
$\frac{N}{2}+2\le m\le N$	Duplicate of values for $2\le m\le \frac{N}{2}$ conjugated and in reverse order so that $〈{\tilde{a}}_m〉={〈{\tilde{a}}_{N-m+2}〉}^{⁎}$

11.5 Measurement as seen from the frequency domain

A complete understanding of the measurement process and its impact on Fourier transform estimates requires that we model that process mathematically. To do this we introduce the convolution theorem. This says that multiplying together two functions in the time domain has the same effect as taking the convolution of (convolving) their Fourier transforms. Conversely, multiplying two transforms is equivalent to convolving their time functions and dividing by 2π. The convolution of two functions a(t) and b(t) denoted as a(t)⁎b(t) is defined as

$c\left(t\right)=a\left(t\right)⁎b\left(t\right)\text{≡}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}a\left(\tau \right)b\left(t-\tau \right)d\tau$

To prove the convolution theorem, we take the Fourier transform of Eq. (11.5.1) over infinite limits.

$\tilde{c}\left(\omega \right)=\frac{1}{2\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}a\left(\tau \right)b\left(t-\tau \right)d\tau e^{i\omega t}dt$

Reversing the order of integration gives

$\tilde{c}\left(\omega \right)=\frac{1}{2\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}a\left(\tau \right){\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}b\left(t-\tau \right)e^{i\omega t}\mathit{dtd\tau }$

With the substitution t′=t−τ we obtain

$\tilde{c}\left(\omega \right)=\frac{1}{2\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}a\left(\tau \right){\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}b\left(t^{\prime }\right)e^{i\omega \left(t^{\prime }+\tau \right)}d{t}^{\prime }d\tau =\frac{1}{2\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}a\left(\tau \right)e^{i\omega \tau }d\tau {\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}b\left(t^{\prime }\right)e^{i\omega t^{\prime }}d{t}^{\prime }=2\pi \tilde{a}\left(\omega \right)\tilde{b}\left(\omega \right)$

So, using the operators [Fscr]{} and ${\mathrm{[Fscr]}}^{-1}\left\{\right\}$ to denote the Fourier transform and its inverse we have that

$\tilde{a}\left(\omega \right)\tilde{b}\left(\omega \right)=\frac{1}{2\pi }\mathrm{[Fscr]}\left\{{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}a\left(\tau \right)b\left(t-\tau \right)d\tau \right\}$

A nearly identical proof shows that

$a\left(t\right)b\left(t\right)={\mathrm{[Fscr]}}^{-1}\left\{{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}\tilde{a}\left(\varpi \right)\tilde{b}\left(\omega -\varpi \right)d\varpi \right\}$

Note that it is simple to show that convolution is commutative, i.e., a(t)⁎b(t)=b(t)⁎a(t) and $\tilde{a}\left(\omega \right)⁎\tilde{b}\left(\omega \right)=\tilde{b}\left(\omega \right)⁎\tilde{a}\left(\omega \right)$. To understand what the convolution operation does, suppose that b is a delta function occurring at a time t₁, i.e., b(t)=δ(t−t₁). Convolving a with b simply shifts a by the time t₁ since

$a\left(t\right)⁎b\left(t\right)=\underset{-\infty }{\overset{\infty }{\int }}a\left(\tau \right)\delta \left(t-t_1-\tau \right)d\tau =a\left(t-t_1\right)$

By extension, convolving a with a sum of two delta functions at t₁ and t₂ produces a function that consists of two copies of a shifted by t₁ and t₂ and added together, i.e., a(t−t₁)+a(t−t₂). In general, we can think of the convolution as smoothing the signal a(t) using the kernel b(t), or vice versa. Exactly analogous examples and interpretation apply to convolution in the frequency domain.

Now consider the measurement process illustrated in Figs. 11.4 and 11.5. We begin with the continuous time variation of a physical quantity a(t) that effectively extends forever, Fig. 11.4A. This signal also has a continuous Fourier transform that in principal extends over an infinite frequency domain, Fig. 11.5A. To measure the signal, we sample it at regular intervals Δt, Fig. 11.4B. This can be modeled as multiplication of the signal by a train of delta functions, one delta function when each sample is taken:

$f\left(t\right)={\displaystyle \sum _{n=-\infty }^{\infty }}\delta \left(t-n\mathrm{\Delta }t\right)\mathrm{\Delta }t$

This has a Fourier transform that is also a train of delta functions since,

$\tilde{f}\left(\omega \right)=\frac{1}{2\pi }{\displaystyle \sum _{n=-\infty }^{\infty }}{\displaystyle \underset{-T}{\overset{T}{\int }}}\delta \left(t-n\mathrm{\Delta }t\right)\mathrm{\Delta }t{e}^{i\omega t}dt=\frac{\mathrm{\Delta }t}{2\pi }{\displaystyle \sum _{n=-\infty }^{\infty }}{e}^{i\omega n\mathrm{\Delta }t}={\displaystyle \sum _{n=-\infty }^{\infty }}\delta \left(\omega -n{\Omega }_o\right)$

where Ω_o=2π/Δt. In the example of Figs. 11.4 and 11.5 the sampling period Δt=3.14 ms, and so Ω_o=2000 rad/s. Thus the effect of sampling is to sum together a series of copies of the Fourier transform $\tilde{a}\left(\mathit{\omega }\right)$ each shifted by a different integer multiple of Ω_o, Fig. 11.6B. The Fourier transform is now periodic with the form

${\tilde{a}}_s\left(\omega \right)={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\tilde{a}\left(\varpi \right){\displaystyle \sum _{n=-\infty }^{\infty }\delta \left(\varpi -\omega +n{\Omega }_o\right)d\varpi =}}{\displaystyle \sum _{n=-\infty }^{\infty }\tilde{a}\left(\omega -n{\Omega }_o\right)}$

This copying is the origin of aliasing. As shown in Fig. 11.5B, for example, $\tilde{a}\left(\mathit{\omega }\right)$ overlaps with the first copy to its right $\tilde{a}\left(\omega -{\Omega }_o\right)$, and the resulting sum produces an aliased curve that is different from both. The overlapping is centered on the Nyquist frequency π/Δt at 1000 rad/s, and the tonal spike at ω=1055 in the original spectrum is aliased down to ω=945. We also see that although aliasing has its greatest effect around the Nyquist frequency, it can influence the entire shape of the Fourier transform. Obviously this overlapping and the associated error do not occur if the original signal has no content above the Nyquist frequency.

The fact that we can only measure the signal for a finite time is equivalent to multiplying it by a function that is 1 while we are measuring and 0 when we are not. This rectangular window function is shown in Fig. 11.4C, and its effect on the time domain data is illustrated in Fig. 11.4D. The window function can be written as

$w\left(t\right)=H\left(t\right)H\left(T_o-t\right)$

where H is the Heaviside step function and T_o is the window length. This has the Fourier transform

$\tilde{w}\left(\omega \right)=\frac{1}{2\pi }{\displaystyle \underset{0}{\overset{T_o}{\int }}}{e}^{i\omega t}dt=\frac{e^{i\omega T_o}-1}{2\pi i\omega }=\frac{T_o}{2\pi }\frac{sin\left(\omega T_o/2\right)}{\omega T_o/2}{e}^{i\omega T_o/2}$

The $\frac{sinx}{x}$, or sinc, function that forms the magnitude of $\tilde{w}\left(\omega \right)$ is dominated by a central peak surrounded by decaying sidelobes with intervening nulls at frequencies of 2πn/T_o, where $\left|n\right|\ge 1$, as illustrated in Fig. 11.5C. So, multiplying the time series by the window function convolves its Fourier transform with this sinc function. The resultant smoothing by $\tilde{w}\left(\omega \right)$, seen in Fig. 11.5D, substantially diminishes the sharp peaks in the transform associated with the tonal components of the signal and also generates sidelobes around those peaks mirroring those present in the window function. This convolution with the window function is the source of broadening.

The final measurement step, implicit in the numerical Fourier transform is the sampling of the Fourier transform in the frequency domain, Fig. 11.5E. The implied multiplication by a delta function train in frequency is equivalent to convolution in the time domain by a delta function train with a spacing equal to T_o, thereby replicating our sampled time signal and making it periodic with a period equal to the window length, Fig. 11.4E.

Unfortunately, we cannot completely mitigate the broadening error since it is a fundamental result of sampling the signal for a finite period of time. The worst effects can be reduced, however, if we window our sampled signal (with its mean value subtracted out) with a function that varies more smoothly to zero at the ends of the window than the default rectangular function of Fig. 11.4C. Numerous options are available, two of which are plotted in Fig. 11.6. These are the Hanning and Blackman Harris window functions which are all given, respectively, by the expressions:

$\begin{array}{l}w\left(t\right)=0.5-0.5cos\left(\frac{2\pi t}{T_o}\right)\hfill \\ w\left(t\right)=0.35875-0.48829cos\left(\frac{2\pi t}{T_o}\right)\hfill \\ +0.14128cos\left(\frac{4\pi t}{T_o}\right)-0.01168cos\left(\frac{6\pi t}{T_o}\right)\hfill \end{array}$

The Fourier transforms of these functions reveal sidelobes that are greatly reduced compared to the rectangular window, though at the expense of increasing the width of the central lobe, Fig. 11.6B. The Hanning window is a satisfactory default choice for most applications. The application of a window reduces the average amplitude of the signal since w(t)≤1. This must be corrected by dividing by the RMS of the window function to ensure that the Fourier transform has the correct level, i.e., by dividing by $\sqrt{\overline{w^2}}$ where

$\overline{w^2}={\displaystyle \underset{0}{\overset{T_o}{\int }}}w{\left(t\right)}^{2}dt$

For the Hanning window $\overline{w^2}=3/8$.

11.7 Wavenumber spectra and spatial correlations

The discussion of spectral estimates that began in Section 11.4 makes exclusive reference to signals in time and the correlations and frequency spectra they imply. However, the analysis results apply equally well to variations over distances, spatial correlations, and wavenumber spectra with minor modifications due to the conjugate relationship between the space and time Fourier transform definitions adopted in Chapter 3. Specifically, the numerical Fourier transform applied to a variation in space along coordinate x₁ sampled at intervals Δx₁ is

$〈{\widetilde{\stackrel{~}{a}}}_m〉=\frac{\mathrm{\Delta }{x}_1}{2\pi }{\displaystyle \sum _{n=1}^N}{a}_n{e}^{-\frac{2\pi i\left(n-1\right)\left(m-1\right)}{N}}=\frac{\mathrm{\Delta }{x}_1}{2\pi }\mathrm{D}\mathrm{F}\mathrm{T}\left(a_n,m\right)$

where a_n=a(nΔx₁), $〈{\widetilde{\stackrel{~}{a}}}_m〉=〈\widetilde{\stackrel{~}{a}}\left(\left[m-1\right]\mathrm{\Delta }{k}_1\right)〉$, and Δk₁=2π/(NΔx₁). Likewise, the inverse transform is

$a_n=\frac{2\pi }{N\mathrm{\Delta }{x}_1}{\displaystyle \sum _{m=1}^N}〈{\widetilde{\stackrel{~}{a}}}_m〉e^{\frac{2\pi i\left(m-1\right)\left(n-1\right)}{N}}=\frac{2\pi }{\mathrm{\Delta }{x}_1}\mathrm{IDFT}\left(〈{\widetilde{\stackrel{~}{a}}}_m〉,n\right)$

Wavenumber spectra are very often calculated by numerical Fourier transform of the associated correlation function, at least when the correlation function is obtained in a point-wise fashion using measurements with two or more probes that are traversed to different positions. However, when an array of sensors is used, or measurements are made optically such as in the PIV snapshot of Fig. 11.2, spectra may be calculated by averaging multiple estimates of the wavenumber transform of instantaneous data sampled in space. In this case, analogs of Eqs. (11.6.3), (11.6.4) are used, specifically

$〈{\varphi }_{\mathit{aa}}^{\left(m\right)}〉=〈{\varphi }_{\mathit{aa}}\left(\left[m-1\right]\mathrm{\Delta }{k}_1\right)〉=\frac{\mathrm{\Delta }{x}_1}{2\pi N{N}_{rec}\overline{w^2}}\sum _{p=1}^{N_{rec}}\mathrm{D}\mathrm{F}{\mathrm{T}}^{⁎}\left(a_n^{\left(p\right)}{w}_n,m\right)\mathrm{D}\mathrm{F}\mathrm{T}\left(a_n^{\left(p\right)}{w}_n,m\right)$

where a_n^(p) denotes the pth record of N samples $a_n=a\left(n\mathrm{\Delta }{x}_1\right)$, and

$〈{\varphi }_{\mathit{ab}}^{\left(m\right)}〉=〈{\varphi }_{\mathit{ab}}\left(\left[m-1\right]\mathrm{\Delta }{k}_1\right)〉=\frac{\mathrm{\Delta }{x}_1}{2\pi N{N}_{rec}\overline{w^2}}\sum _{p=1}^{N_{rec}}\mathrm{D}\mathrm{F}{\mathrm{T}}^{⁎}\left(a_n^{\left(p\right)}{w}_n,m\right)\mathrm{D}\mathrm{F}\mathrm{T}\left(b_n^{\left(p\right)}{w}_n,m\right)$