A Few Useful DFT Properties and Theorems

These are useM as practical tools, and for gaining intuition about the time and frequency domains. I ll prove some of them, and leave the rest for you. 

The only difference between the forward (time to frequency) and inverse (frequency to time) DFT is a negative sign in the sinusoidal exponent, and the MNnormalization lactor that is applied in one direction. This implies that any transform property can apply in either direction, sometimes with the appropriate correction for the minus sign and normalization. 

This proof assumes that either the signal buffer is zero-padded by at least P samples in the direction of the shift, so that we don t lose any samples when we perform the shift. Alternatively, the shift could take place circularly within the buffer (negative P wraps to A- P in the buffer, etc.). The trick in doing the proof is to exploit the properties of exponents where addition within the exponent is equivalent to multiplication of two exponents with a common base e e = e ). 

The implication of the shift theorem is that all that happens to the transform of a time-shifted signal is a phase shift in the frequency domain, linear in m with slope equal to the amoimt of shift, represented by the Nothing 

To imderstand the shift theorem in the frequency domain, we can invoke the duality of the time and frequency domains as discussed in Section A.5.1. Instead of saying that we multiplied the sine wave by the window, we could say that we multiplied the window by the sine wave. Doing this shifted the transform of the window from zero frequency up to the frequency of the sinusoid. This is sometimes called heterodyning, or shifting a spectrum by multiplication by a sinusoid in the time domain. 

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