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Remember, the Mk are twice the |Ck|, but the angles are the same for both. We've also seen how these Ck build a good approximation to the desired waveform (Fig ç ), so we can start to gain confidence in this new way of working with signals. But, before we leave it, we want to formally define the "Transform" method that we've uncovered. We also want to "stand back" from all the details, so that we can appreciate the full significance of this result. 3 The DfFT (or Fourier Series) We've just described a transform mechanism that is commonly known as Fourier Series.
This plot (Fig ê ) shows both. 57 radians). This means that sin(ωt) = cos(ωt − π/2). 30 New Page 1 We've called the cosine wave xe(t) because of its even symmetry about the vertical zero−axis. We've called the sine wave xo(t) because of its odd symmetry about the same axis. These symmetries are defined by saying: A given (real−valued) x(t), such as this one (Fig ç ), might have neither symmetry, but we can routinely decompose it into even and odd parts. Its even part is: 31 New Page 1 Notice, the right−hand−side (rhs) of the equation is unchanged if we use −t in place of +t..
Relative to one another, they are stationary. That's why we can add them all as vectors to get this result. In strict maths notation, we must replace ∠θ by ejθ everywhere. This is the method that we use in AC circuit theory. We can treat all voltages and currents as vectors, because they all have the same frequency, and when they add they retain their sinusoidal shape. You can't say that about other waveforms (square, triangle, etc). Moving on from just one frequency, we will allow different frequencies fk, but with this constraint: The permitted frequencies are integer multiples of ε, where ε is the fundamental frequency.