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4
Data Acquisition and Analysis

This chapter fundamentally deals with how we'll digitally represent the various aircraft performance characteristics for further analysis. A critical element of this chapter is a detailed discussion of how digital data acquisition (DAQ) systems work. Our discussion of DAQ techniques is motivated by industry and military flight test programs, which rely on complex data acquisition systems. Instrumentation on board the aircraft can involve hundreds or thousands of sensors of various kinds, and an equivalent number of channels to digitize and record this data. Data can be sampled at high rates in order to effectively capture transient phenomena, leading to vast quantities of data which require high bandwidth and storage. Furthermore, these data are often streamed in real time to ground stations via radio telemetry link, such that flight test engineers can monitor the data as the test is being conducted. Live telemetry of data and real‐time analysis adds to the safety and efficiency of the flight test program, enabling the flight test team to avoid hazardous test conditions or to adapt to test events as they develop.

While the typical student will not be able to work with such high‐end systems in the university environment, the basic principles of DAQ are still relevant in low‐cost, small‐scale data acquisition systems. With the continued evolution and miniaturization of digital electronics, these systems are now readily accessible even to students. Simple data acquisition systems in contemporary flight testing use include smartphones (Gregory and Jensen 2012; Gregory and McCrink 2016), LabVIEW‐ or MATLAB‐based digital DAQ systems (Muratore 2012), Arduino microprocessors (Koeberle et al. 2019), commercially available systems, and even the standard avionics onboard the aircraft (see Chapter 3). Thus, students can readily get exposure to the basic principles of DAQ systems, methods, and data analysis employed in flight testing.

This chapter provides an overview of the foundations related to DAQ and processing, such that the capabilities and limitations of DAQ methods can be appreciated. Our discussion of DAQ will begin with defining how signals may be represented as a function of time or frequency. This will directly lead to a discussion of filtering, whereby unwanted frequency content can be attenuated in a signal. Following this, we'll focus on the essential characteristics of DAQ, with an overview of the methods used to digitally represent an analog signal. We'll then conclude with an example of how DAQ techniques are applied to flight testing.

4.1 Temporal and Spectral Analysis

We'll start our discussion of DAQ by considering how signals can be represented in either the time domain or the frequency domain. Both representations of a signal are of use in the flight testing environment. The time domain representation is the most intuitive, where the signal is plotted as a function of time. The frequency domain representation is less intuitive, but no less powerful. This form of presenting a signal shows the relative significance of different frequency components found in the signal, facilitating visual separation of different frequency peaks in the spectrum. We'll examine both approaches of data representation as follows, and establish the link between the two.

Figure 4.1 Sample signal in the time domain.

A time‐domain representation of a signal is our intuitive view of signal waveforms, which is a plot of voltage as a function of time. For example, Figure 4.1 shows a plot of the function

(4.1)

in the time domain, where c₀ = 2, c₁ = 1, c₂ = 0.5, c₃ = 0.5, ω₁ = 50.3 rad/s (f₁ = ω₁/2π = 8 Hz), ω₂ = 5ω₁, and ω₃ = 10ω₁. A representation of the same signal in the frequency domain, however, will plot the amplitude of each frequency component of the signal versus the corresponding frequency. Inspection of Eq. (4.1) reveals that the signal has three non‐zero frequency components: a dominant, low‐frequency component with amplitude of c₁ at a frequency of ω₁; and two smaller, higher‐frequency components with amplitudes c₂ and c₃ at frequencies ω₂ and ω₃. We can take these three amplitudes (c₁, c₂, and c₃), along with the time‐averaged amplitude of the signal (c₀, which has a frequency of ω₀ = 0) and plot them versus the corresponding frequencies, resulting in the frequency domain plot shown in Figure 4.2.

Plotting the signal represented in Figure 4.1 in the frequency domain is straightforward in this case since we know the amplitude and frequency of each component from Eq. (4.1). But what if we don't know the frequency content of a signal? How would we generate a frequency domain representation of some arbitrary signal that we have acquired? The answer lies in a computational technique known as the fast Fourier transform (FFT), which is a straightforward and fast algorithm for computing spectral content of digitized signals. This section will provide only a very abbreviated overview of spectral analysis, which is the process of representing a time‐based signal in the frequency domain. The interested reader is encouraged to consult other sources such as Wheeler and Ganji (2003) or Bendat and Piersol (2010) for further details.

The concept behind Fourier analysis is that an arbitrary, periodic signal may be represented by a summation of a constant with an arbitrary number of sine and cosine functions (Powers 1999). The mathematical representation of a Fourier series is given as

(4.2)

Figure 4.2 Sample signal in the frequency domain.

Note that our example function defined earlier (Eq. (4.1)) is of the same form as Eq. (4.2), which made it straightforward for us to pick out the coefficients of the Fourier series by inspection and plot the signal in the frequency domain (Figure 4.2). For an arbitrary periodic waveform, the coefficients in Eq. (4.2) may be determined from

(4.3)

which is the time‐average of the signal over one period of the waveform, T = 1/f = 2π/ω. The other coefficients are given by

(4.4)

Note that Eq. (4.2) is an infinite series, implying that an infinite number of coefficients may be required to fully represent an arbitrary periodic waveform. In practice, the number of coefficients used to represent a signal is truncated to some reasonable number of computationally determined coefficients. If Eqs. (4.2)–(4.4) are evaluated for the sample function given by Eq. (4.1), with T = 2π/ω₁ = 0.125 seconds, we could directly compute the integrals and find that the coefficients of the Fourier series are a₀ = c₀, b₁ = c₁, b₅ = c₂, and b₁₀ = c₃, with all other coefficients being zero.

Figure 4.3 shows another example of the application of a Fourier series representation to a triangle waveform,

(4.5)

Figure 4.3 Fourier series approximation of a triangle waveform.

with frequency ω = 314.2 rad/s (f = 50 Hz). Note that the function represented in Eq. (4.5) is odd, meaning that f(t) =  − f(−t). The implication of this for the Fourier series is that the a_n coefficients are zero and the function can be represented entirely by the sine terms in Eq. (4.2), making it a Fourier sine series. The coefficients of this sine series are b₁ =  − 0.8106, b₃ = 0.09006, b₅ =  − 0.03242, b₇ = 0.01654, …. Inclusion of each successive term in the sine series improves the fidelity of the Fourier approximation to the original waveform (Figure 4.3), with higher frequency coefficients improving the fit at the peaks of the triangle waveform. This is because the amplitude of the higher order terms decays rapidly with frequency, as shown in Figure 4.4.

Figure 4.4 Fourier components of the sine series approximation to a triangle waveform.

While the definition of the Fourier series is useful for illustrating the representation of signals in the frequency domain, the development thus far is not yet useful for frequency representation of an arbitrary waveform that we might encounter in flight testing. Actual analysis of signals via Fourier techniques is done through the Fourier transform, which relaxes the constraint on periodicity. The steps in frequency (ω_n = nω) in Eq. (4.2) are reduced until ω becomes a continuous function of frequency and we have

(4.6)

which is the Fourier transform of a function f(t). Here the exponential function is another way of representing the sine and cosine terms, and such that F(ω) is a complex‐valued function (see Wheeler and Ganji 2003 or Bendat and Piersol 2010 for further details). Similarly, if one has a defined Fourier transform, F(ω), the original signal f(t) can be determined from the inverse Fourier transform,

(4.7)

In applying the Fourier transform to a digital representation of a signal (sampled at discrete time intervals) the discrete Fourier transform (DFT) is used. The DFT is represented by

(4.8)

where N is the total number of samples in the data record, Δt is the time interval between samples, Δf is the increment in frequency (equal to the inverse of the period), and there are k frequency components in the transform. (Note that only values up to k = N/2 are unique). In the same way that we defined an inverse Fourier transform, we can also define an inverse DFT,

(4.9)

for recovering the original signal.

The DFT and inverse DFT can be determined numerically, but this process tends to be computationally expensive since the number of computations is on the order of N² real‐valued multiply‐add operations. Due to this computational expense, the FFT technique has been developed, which requires on the order of Nlog₂N computations. (For a record length of N = 2¹⁶, the FFT requires 2¹² fewer computations compared to the DFT). Bendat and Piersol (2010) may be consulted for complete details on the derivation and implementation of the FFT algorithm. Returning to our analysis of a triangle waveform (Eq. (4.5)), spectral analysis based on the FFT algorithm results in the power spectrum shown in Figure 4.5. The dominant frequency and higher harmonics are faithfully captured by the FFT algorithm, as is evident when we compare Figure 4.5 with Figure 4.4.

Note that the power spectral density is often represented on a logarithmic scale, which helps the higher‐frequency components of the signal appear more prominently on the plot (this is in contrast to the linear scaling employed in Figures 4.2 and 4.4). Power spectra often have broader frequency peaks along with lower‐amplitude ripple between the peaks, which are characteristic of the DFT and FFT algorithms. This phenomenon, referred to as spectral leakage, results from a non‐integer number of waveforms being present within the data record analyzed by the FFT, with the end effects being the primary culprit. Leakage may be reduced through windowing, where the window is a tapering function (high in the center region, with low values at the ends) multiplied with a subset of the data. Since the window size is a subset of the full data record, the full FFT is then the average of the computed FFTs of the windowed portions of the signal. An amount of overlap may be specified, which determines how much the window is shifted along the length of the record. Welch's modified periodogram method (Welch 1967) is one common approach to implementing a windowing function with overlap (see the function in MATLAB). Welch's method reduces the noise in the power spectrum, at the expense of reduced frequency resolution.

Figure 4.5 Power spectrum based on the FFT of the triangle waveform.

4.2 Filtering

We'll now discuss filtering techniques, and how they can be used to improve the interpretation of the signal by removing unwanted frequency content. When considering a signal represented in the frequency domain, filtering suppresses the amplitude of signal content over a select range of frequencies. This is done by defining and applying a transfer function, which is a frequency‐dependent weighting value that is multiplied with the signal in the frequency domain.

Common types of filters include low pass, high pass, band pass, and band stop schemes, which are illustrated in Figure 4.6. A low pass filter will attenuate signal content at frequencies above a specified cutoff frequency, and preserve signal content below that cutoff frequency. Low pass filtering is useful for removing high‐frequency noise in a signal, which may obscure the desired low frequency data. High pass filtering does just the opposite – it attenuates signal content at frequencies below the cutoff, and preserves signal content at higher frequencies. High pass filtering is particularly useful for removing the steady‐state voltage (a DC mean value) from a signal. The band pass filter is essentially a combination of the two, where the high‐pass cutoff frequency is at a lower frequency than the cutoff for the low‐pass filter. The region between the two cutoff frequencies is the passband. Finally, a band stop filter is designed to selectively remove a range of frequencies – it is the logical inverse of a band pass filter, where the low‐pass cutoff frequency is below the high‐pass cutoff frequency.

Figure 4.6 illustrates the effect of each of these filters on a signal (Eq. (4.1)) in both the time domain (left column) and in the frequency domain (center column). The right‐most column represents the transfer function for each filter. A filter can be applied to a given signal by transforming that signal into the frequency domain (via FFT), linearly multiplying the signal's power spectrum with the filter's frequency response function in the frequency domain, and then transforming the signal back into the time domain (via the inverse FFT). In Figure 4.6, the baseline signal is the same waveform that we considered in an earlier example (see Eq. (4.1)). The effects of the low pass filter are to suppress the higher frequency components at 40 and 80 Hz, leaving only the DC level and the sinusoidal component at 8 Hz. In contrast, the high pass filter removes the DC level and the lowest frequency component (8 Hz), leaving both high frequency components unattenuated. The band pass and band stop filters have equivalent performance, where the filter suppresses signal content in portions of the spectrum where the filter's transfer function has a high level of attenuation.

Figure 4.6 Examples of various filtering schemes applied to signals in the time domain (left column) and frequency domain (center column). The right column shows the transfer function associated with each defined filter: low pass, high pass, band pass, and band stop.

Beyond the definition of the various filter types, there are other filter characteristics that are important to consider. Common filter classes include Butterworth, Bessel, Chebyshev, and elliptical – each of which has varying characteristics. One important parameter is the filter order, which governs the stopband attenuation rate – a description of how much increase in attenuation can be achieved over a given frequency interval. Attenuation rates are typically specified as dB/octave or dB/decade, where an octave is a factor of two change in frequency and a decade is an order of magnitude change in frequency. Filter order is related to attenuation rate for the Butterworth filter by 6m dB/octave, where m is the filter order. A third important parameter is the amount of ripple allowed in the passband or the stopband. Figure 4.6 illustrates ripple throughout the stopband, where the transfer function is not flat across a range of frequencies. There is typically a tradeoff between ripple and attenuation rate, where high attenuation rate is achieved at the expense of increased ripple, and the filter class has a significant impact on the attenuation rate.

One final, critical characteristic of filters is the phase lag induced by the filter. In the same way that a filter exhibits attenuation as a function of frequency, there will be induced phase delay that is also frequency‐dependent. In most data processing applications this is an undesirable feature of the filter, but can be worked around through careful filter design or creative application of the filter (e.g., the function in MATLAB's Signal Processing Toolbox, which feeds a signal forward and then backward through the filter in order to cancel the phase effect). Phase delay in processed flight test data can be important when comparing a filtered signal with an unfiltered signal. A detailed discussion of filter design is beyond the scope of this text, but appropriate resources may be consulted for further details (Wheeler and Ganji 2003; Bendat and Piersol 2010).

Filtering can be applied before digitization or afterwards. Typically pre‐DAQ analog filtering is used to prevent aliasing, which will be discussed in Section 4.4. Analog filtering involves the use of dedicated circuitry, and once the signal has been digitized there is no longer any flexibility to change the filter cutoff frequency or characteristics. Digital filtering, on the other hand, can be changed at will during post‐processing, allowing an interactive and adaptive approach to data analysis.

4.3 Digital Sampling: Bit Depth Resolution and Sample Rate

Let's now consider the details of how a signal is actually captured in digital form. The fundamental principle of DAQ is creating a digital representation of an analog signal. An analog signal is defined as one where the signal level (e.g., voltage) is a continuous function of time. Digital signals, however, are always a discretized representation of that continuous function, with the fidelity of that representation depending on how many discretization levels are used across amplitude and time. Resolution of the signal amplitude depends on the bit‐depth resolution of the data acquisition device, and the defined input range. The input range essentially determines the minimum and maximum voltages that can be recorded for a given measurement, where any input values exceeding those limits will be clipped. Typical data acquisition ranges can be unipolar, where all of the input voltages are of the same sign (e.g., 0–5 V, or 0–10 V), or bipolar, where positive or negative voltages can be measured (e.g., −5 to 5 V, or −10 to 10 V). Bit depth resolution is a measure of how many discretization levels are used to subdivide the input range. This resolution is typically expressed as a power of 2, due to the architecture of the data acquisition hardware. For example, a 12‐bit data acquisition device will have 2¹² (4096) discretization levels spanning the input range.

A combination of input range and the bit depth resolution defines the minimum change in voltage that can be resolved in a digital waveform. The maximum error between a given analog voltage and its digital representation is the quantization error,

(4.10)

where R is the input range, and B is the number of bits of the DAQ device. Digitization of a desired signal should use an input range that is as close to the limits of the anticipated signal as possible (with little risk of the signal exceeding that input range) and a bit depth resolution as high as possible. The downsides of increased bit depth resolution are increased cost of the data acquisition hardware, and larger file sizes required to store the digitized signals. In practice, the bit depth resolution should be sufficiently high such that the discretization error is small relative to the smallest voltage change in the desired signal.

Figures 4.7 and 4.8 illustrate the effects of bit depth resolution on the digital representation of an analog signal. For this example, a simple sine wave with frequency of 1000 Hz (628 rad/s) is defined as

(4.11)

A digital representation of that signal with an input range of 0–5 V and a 4‐bit converter (2⁴, or 16 steps) is shown in Figure 4.7, compared to the original analog function. The step‐stair appearance of the signal is due to quantization error, where the digital representation of the continuous waveform is rounded off to the nearest quantization level. If the bit depth resolution is increased to 2¹², as shown in the zoomed‐in waveform in Figure 4.8, the signal representation is much more faithful to the original analog signal. The increase in bit depth resolution from 2⁴ to 2¹² provides a factor of 256 more levels to represent the analog waveform than the 4‐bit case shown in Figure 4.7.

Sample rate, defined as the number of digital samples acquired per second, is the other predominant factor that dictates the fidelity of the digital representation of the analog waveform. Sampling rate is determined by the time required to perform the analog‐to‐digital conversion process, limiting how many samples can be digitized in a given amount of time. If the sample rate is low – i.e., there is a long period of time between each sample of the analog signal – the acquisition process could miss important changes in the analog signal in the intervening time. Not only is important information missed by an insufficient sample rate, but the resulting digital representation can be misleading.

Figure 4.7 Digital representation of an analog waveform with input range of 0–5 V, a 4‐bit converter, and very high sample rate.

Figure 4.8 Digital representation of the analog waveform with input range of 0–5 V, a 12‐bit converter, and very high sample rate.

The sample rate must be sufficiently high relative to the highest frequency present in the signal. Note that signals can be a superposition of many constituent frequencies. As a general rule of thumb, a sample rate of at least 10 to 20 times the highest frequency present in the signal will provide a reasonable representation of the analog signal in the time domain. For example, Figure 4.9 shows a generally insufficient representation of the analog signal, where the sample rate of 4400 Hz is only 4.4 times higher than the 1000 Hz frequency being measured. In contrast, Figure 4.10 shows a fairly good representation of the signal, with a sample rate of 22 000 Hz (22 times the frequency of the sampled waveform), with only minor residual error in capturing the magnitude of the waveform peaks (approximately 80 mV error).

Figure 4.9 Digital representation of the analog waveform with a sample rate of 4400 Hz (4.4 samples per cycle of the 1000 Hz analog waveform), with very high bit depth resolution.

Figure 4.10 Digital representation of the analog waveform with a sample rate of 22 000 Hz (22 samples per cycle of the 1000 Hz analog waveform), with very high bit depth resolution.