1. 7.2 Even and Odd Fourier Transforms phase of signal frequencies
breaking a function into even and odd components using calculations
breaking a function into even and odd components using graphs
representation of even and odd functions in the complex plane
spectrum of a train of rectangular pulses
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2. Phase of Signal Frequencies
A signal is composed of frequencies each of which has a specific phase with respect to time zero.
By making the spectrum consist of both sines and cosines, non-zero phases can be represented. For example, the function
can be written as a cosine with a 45° phase lag.
Even temporal functions have a spectrum composed of only cosines (0° phase lag, even function). Odd temporal functions have a spectrum composed of only sines (90° phase lag, odd function).
In order to use a graphical approach to determining the Fourier transform it is necessary to decompose the signal into even and odd functions.
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3. Obtaining Even & Odd Functions (1)
An even function is defined as one where fe(-t) = fe(+t). An odd function is defined as one where fo(-t) = -fo(+t).
Any function can be written as a sum of even and odd terms.
The two equations above can be solved for the even and odd functions.
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4. Obtaining Even & Odd Functions (2)
One of the goals of this section of the course is to use the basis set transform pictures to obtain a graphic solution to the Fourier transform. Thus it would be convenient to learn how to graphically decompose a picture into even and odd parts.
Consider the triangularly-shaped temporal waveform shown at the left below.
To obtain the even waveform, first mirror image the blue from quadrant I to quadrant II to obtain the red. Then add the two waveforms together.
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5. Obtaining Even & Odd Functions (3)
To obtain the odd picture, mirror image the blue waveform from quadrant I to quadrant II, and then to quadrant III. Add the blue and red waveforms together.
The two pictures need to be scaled so that their sum equals the starting picture. This is accomplished by multiplying each by 1/2. The amplitude on the left sums to zero, while that on the right sums to the original waveform.
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6. Using the Complex Plane
When an even temporal function is transformed, it produces a spectrum composed of only cosines. An odd function produces a spectrum of only sines.
The most useful representation of the combined data uses the complex plane where the amplitude, R, and phase, q, are given by the equations to the right.
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7. Spectrum of Rectangular Pulses
The inter-pulse spacing is 100 s, dictating that all frequencies are multiples of 0.01 Hz. Since the integral is non-zero, a dc component exists. The spectrum is composed of sines and cosines since the signal is neither even nor odd.
The pulse train shown has 20 frequencies. Their amplitude and phase are shown below. The fifth harmonic is 0.05 Hz, etc.
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