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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.. This proves that xe(−t) = x e(t) as per the definition. The odd part of x(t) is found as: Notice, the rhs of this equation is sign−reversed if we use −t in place of +t.

From the symmetry of these waveforms, and the symmetry of the CFT/ICFT integrals, it also emerges that the CFT of a sinc in time is a rectangle−shaped spectrum, that is: To interpret this result: because the sinc−shaped time−pulse is so smooth, its spectrum is band−limited. It has no frequencies higher than ½. These CFT pairs have great significance, and we will use them shortly.

Its of some interest to see how a sine wave would respond. We'll take time reversal first. Suppose x(t) = cos(2πf 1t+α). Then x(−t) = cos(2πf 1(−t)+α) which is the same as saying x(−t) = cos(2πf 1t−α) because the cos is even (cos(−θ) = cosθ). The effect of time reversal is to change the sign of α. This plot offers graphical confirmation of that idea (Fig ë ). 36 New Page 1 To delay a sine wave x(t) by τ seconds, we must wind back both phasors by a corresponding angle, going against their direction of rotation (Fig ç ).

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DSP Lectures

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