fft3 - SIGNAL PROCESSING & SIMULATION NEWSLETTER...

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Discrete Fourier Transform (DFT) and the FFT Let’s take this continuos signal composed of three sinusoids. Assume that f a = 1, f b = 3, and f c = 5. The waveform is plotted below. Figure 1 - A signal representing a square wave We can draw the Fourier transform of this signal easily by examining the amplitudes of each of these frequencies and then putting one-half on each side of the y-axis as shown in Fig. 2 for a two sided spectrum. Real signals such as this one produce only one sided spectrums also shown below.
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Figure 2 - The two-sided and single-sided Fourier Transform of g(t) This is the theoretical Fourier Transform of the continuos waveform g(t). The Fourier Transform tells us that there are just three frequencies in the signal and no others. There is no ambiguity in the results. This ideal Fourier Transform is what we want to see when do the Fourier Transform on a analyzer or on a computer but in reality this is nearly impossible to obtain. All implementations of the Fourier Transform are attempts to achieve the theoretical results, however, digital signal processing introduces approximations and truncation effects which keep us from realizing the ideal. Figure 3 shows the outline of the same signal along with dots that represents what we actually see of the signal on a oscilloscope. This is because most signals we capture are sampled versions of the real analog signal. We pulse the analog signal every so often and then plot these sampled values. We connect the samples and get a proxy to the signal. Figure 3 - The discrete samples of a real signal as shown by dots. We really do not know the underlying shape of the signal. Mathematically, the sampled signal is obtained by multiplying the target signal with an impulse train of period τ . Since we usually collect only a limited number of samples we limit the length of the impulse train to a certain time window. The discrete signal is expressed as
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So before we can even look at a signal, two things have happened. 1. we have multiplied the target signal by an impulse train and made the continuos signal a discrete signal and 2. we have chosen to collect only a limited number of samples, in effect windowing the sequence with a rectangular window function. Fig 4a shows the original signal and its Fourier Transform. In (b) we have the Fourier Transform of a pulse train which is used for sampling the original signal. The Fourier transform of the impulse train consists of just one frequency, the sampling frequency. Next step is the rectangular window that limits the infinite impulse train. Its Fourier Transform is shown in c and is the well-known sinc function. Now we have a signal which is a product of three signals.
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fft3 - SIGNAL PROCESSING & SIMULATION NEWSLETTER...

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