1. MECH 373 Instrumentation and Measurements
Lecture 11
Discrete Sampling & Analysis of Time-Varying Signals (Chapter 5)
• Sampling-Rate Theorem (review)
• Spectral Analysis of Time-Varying Signal
• Spectral Analysis uisng the Fourier Transform

2. Spectral Analysis
Example 2: Find the Fourier coefficient of the periodic function

3. Spectral Analysis

4. Spectral Analysis

5. Spectral Analysis Using Fourier Transform
• As discussed previously, most of the functions can be represented by the sum of a set of sine and cosine functions (Fourier series).
• However, the technique most commonly used to spectrally decompose functions is the Fourier Transform.
• The Fourier transform is a generalization of the Fourier series.
• The Fourier transform can be applied to any practical function and does not require that the function be periodic.
• Although the function f(t) is considered to be continuous for the Fourier series analysis, the Fourier transform can be applied to the discrete data, which can be evaluated quickly using modern computer technique called the Fast Fourier Transform or FFT.
• In presenting Fourier transform, it is common to start with Fourier series, but in a different form. This form is called the complex exponential form.
• The sine and cosine functions can be represented in terms of the complex exponentials as

6. Spectral Analysis Using Fourier Transform
• In the previous lecture we discussed that a portion of a non-periodic function can be represented by a Fourier series by assuming that the portion of duration T is repeated periodically.
• The fundamental angular frequency, ω0 is determined by this selected portion of the signal as, ω0 =2π/T.

7. Spectral Analysis Using Fourier Transform
• If a longer value of T is selected, the lowest frequency will be reduced.
• This concept can be extended to make T approach infinity and the lowest frequency approach zero. In this case, frequency becomes a continuous function.
• It is this approach that leads to the concept of Fourier transform.
• The Fourier transform of a function f(t) is defined as:
where, F(ω) is a continuous complex-valued function.
• Once a Fourier transform has been determined, the original function f(t) can be recovered from the inverse Fourier transform, that is
• In experiments, a signal is measured only over a finite time period, and with computerized data-acquisition systems, it is measured only at discrete times.

8. Spectral Analysis Using Fourier Transform
• Such a signal is not well suited for analysis by the continuous Fourier transform.
• For data taken at discrete times over a finite time interval, the discrete Fourier transform (DFT) has been defined as
where, N is the number of samples taken during the time period T.
• The increment of f, Δf, is equal to 1/T, and the increment of time (the sampling period Δt) is equal to T/N.
• The amplitude of F for a given frequency represents the relative contribution of that frequency to the original signal.
• The sampling rate is N/T, so the maximum allowable frequency in the sampled signal will be less than one-half of this value, or N/2T = NΔf/2.

9. Spectral Analysis Using Fourier Transform
• The original signal can also be recovered from the discrete Fourier transform using the inverse discrete Fourier transform, given by
• The discrete Fourier transform can be performed using computers.
• The amount of computer time required is roughly proportional to For large values of N, this can be prohibitive.
• A sophisticated algorithm called Fast Fourier Transform (FFT) has been developed to compute discrete Fourier transforms much more rapidly.
• The only restriction for FFT is that the value of N should be a power of 2. For example, 128, 256, 512, 1024 and so on.
• The fast Fourier transform algorithm is also built into devices called spectral analyzers, which can discretize an analog signal and use the FFT to determine the frequencies.

10. Spectral Analysis Using Fourier Transform
• To examine some of the characteristics of the discrete Fourier transform, consider a function that has 10 Hz and 15 Hz components. That is,
• Since this function is composed of two sine waves, with frequencies of 10 Hz and 15 Hz, we would expect to see large values of F at these frequencies in the DFT analysis of the signal.
• The effect of the number of discrete data points within a given time period is shown below.

11. Spectral Analysis Using Fourier Transform
• If we discretize one second of signal into 128 samples and perform an FFT, the results is shown in Fig
• The figures show that as expected, the magnitudes of F at f = 10 Hz and f = 15 Hz are dominant.

12. Spectral Analysis Using Fourier Transform
• The comparison of the two figures show that this effect is reduced when large number of data points within the given time period are considered.
• For the FFTs shown in the above figures, the data samples contained integral numbers of complete cycles of both component sinusoids. That is, integral number of complete waves.
• In general, the experimenter does not know the spectral composition of the signal and will not be able to select the sampling time T such that there will be an integral number of cycles of any frequency in the signal. This complicates the process of Fourier decomposition.
• To demonstrate this point, the lowest frequency is modified from 10 Hz to Hz. That is
• Now in one second period, integer number of complete cycles for 15 Hz component are present but for Hz component, the last cycle is not completed and thus, the complete cycles is not an integer number. This effect is show in the following figure.

13. Spectral Analysis Using Fourier Transform
• The figure shows that this effect (i.e. the non-integral number of cycles) has altered the entire spectrum.
• These significant amplitudes at frequencies not in the original signal are caused by an effect called leakage.
• A common method used to reduce leakage is known as windowing.
• Since leakage is caused by incomplete cycles at the ends of the sampling period, windowing seeks to minimize the end effects by multiplying the signal by a weight function that is larger at the center of the signal sampling period than it is at the ends.