School of Aerospace Engineering Copyright  1999-2002, 2005 by J. Seitzman. All rights reserved. 1 AE3051 Experimental Fluid Dynamics DIGITAL SAMPLING OF TIME-DEPENDENT SIGNALS Objective The primary objective of this experiment is to familiarize the student with digital data acquisition of time-varying signals. This lab covers concepts in frequency analysis of time-varying signals and sampling theory. It also provides an introduction to computer-based data acquisition systems. In this experiment, you will use a computer data acquisition system to sample signals produced by waveforms stored on a compact disk (CD) and converted to an analog electrical signal by a CD player. You will explore issues in sampling, including the Nyquist limit and aliasing, and the use of analog filters. This will help prepare you for future experiments in this laboratory course that employ computerized data acquisition and involve frequency based interpretation of measured data. Background Most experimental measurements involve the dimension of time. Experimental data is acquired over the course of some time, and during this time the signal can change. In some cases, the actual physical parameter of interest (the measurand) may change with time. For example, the velocity in a wind tunnel generally varies with time due to turbulence or variations in the speed of the fan blades used to drive the tunnel. Even when the measurand is nominally constant in time, other parameters that influence the measurement may vary, for example drifts in the measurement device. Thus, the experimenter is often interested in measuring a variable that could be described by the general function (or waveform), v=v(t). (1) a) Waveforms, Frequency Content and Discrete Sampling Fourier Series One of the simplest time-dependent functions we encounter is the sine (or cosine*),       nftAtAt 2sinsinv (2) *Either function is acceptable, since sin(t)=cos(t-/2), i.e., the two functions are identical except for a phase difference of /2 or 90, meaning that shifted by one-fourth of a cycle, cosine looks just like sine.AE 3051 Digital Sampling of Time-Dependent Signals 2 where A is the amplitude,  is the circular frequency (e.g., rad/s), f is the cyclic frequency (e.g., cycles/s, Hertz or s-1), and  is the phase, which represents the time-shift of the sine-wave from some reference time that defines t=0. Such a function is often denoted as a simple harmonic waveform. More general periodic waveforms, which repeat themselves with a period T and thus have a frequency f=1/T, can be written as a linear combination of simple harmonic modes. There is the basic, fundamental mode (with frequency f), and harmonics of the fundamental mode, with integer multiples of its frequency (2f, 3f, …). For example, we could describe the vibrations of a tuning fork or the acoustic oscillations in a pipe this way. Mathematically, this linear combination of modes is expressed as a Fourier series expansion,       102sin2cosvnnnnftbnftaat (3) where nf represents the frequency of the nth mode (n=1 for the fundamental, n=2 for the first harmonic, etc.), a0 represents the steady component of the waveform, and the an, bn are the harmonic coefficients (or amplitudes) of each mode. The steady amplitude, a0, is often called the DC component of the waveform, in reference to classical electrical power systems, which are either Direct Current (steady) or Alternating Current (sinusoidal with a zero average). For example, Figure 1 shows a simple waveform composed of two frequencies, a fundamental mode at 50 Hz and its 9th harmonic (at 500 Hz). Thus the complete waveform is repeated every 20 ms (=1/fundamental frequency =1/50 s). The waveform shown in the figure also has a DC component. In other words, the signal has a nonzero value when averaged over its period. In general, we can write the DC amplitude as    22220vv1TTTTdttfdttTa. (4) The other coefficients of the Fourier expansion are given by       22222sinv22cosv2TTnTTndtnfttfbdtnfttfa , (5)AE 3051 Digital Sampling of Time-Dependent Signals 3 and they can be combined into a complex number,  222v2TTnftinndtetfbia . (6) The power, P, contained in single mode is given by the square of the amplitude 22nnba)n(P  (7) and the phase  (or phase angle) of a mode is given by  nnabtann1. (8) A second example that shows the ability of a combination of sine waves to create an arbitrary periodic function is shown in Fig. 2. Five sine waves and a DC component (see Fig. 3) were combined to create a function approaching a square wave. While the constructed function resembles a square wave, it is clear that more sine waves would be needed to produce a sharp square wave. Fourier Transforms The procedure outlined above for periodic functions can be extended to general functions, which are not necessarily periodic, by considering any arbitrary function to be periodic with an infinitely long period. This approach leads to the Fourier Transform. Given a function v(t), its Fourier Transform V(f) is a complex function defined by     dtetffti2vV (9) in parallel to the complex Fourier function of equation (6). The function V(f) represents the information given by v(t) transformed from the time domain to the frequency domain. The transformation is nearly identical in the reverse direction, with simply a change in the phase (note the sign of the exponent), i.e.,     dfeftfti2Vv . (10) Since, e-ix=cosx - i sinx.AE 3051 Digital Sampling of Time-Dependent Signals 4 For example, Figure 4 graphically shows the Fourier transforms of various functions, including sine and cosine waves, a rectangle function (), a triangle function () and a constant, or DC, function. The sine, cosine and DC waveforms result in Fourier transforms that are nonzero at a single frequency*; in other words, they contain information at only one frequency (the DC function, which does not change in time, is associated with a frequency of zero). The Fourier transforms of the rectangle and