The Fourier Transform

Continuous Time Fourier Transform

Definition 24

The Continuous Time Fourier Transform converts an aperiodic signal into the frequency domain.

Definition 25

Properties of the CTFT

Time Shift:

Time/Frequency Scaling:

Conjugation:

Derivative:

Convolution/Multiplication:

Frequency Shift:

Parsevals Theorem:

Convergence of the CTFT

Theorem 5

Conceptually, this theorem makes sense because

Now if we apply the frequency shift property, we see that

With this, we can define our generalized Fourier Transform for periodic signals.

Definition 26

This definition works because any periodic signal can be represented by its Fourier Series. The rational behind using the Dirac Delta in this generalized Fourier Transform is exlained by the Theory of Distributions which can be found in Appendix.

Discrete Time Fourier Transform

Definition 27

The Discrete Time Fourier Transform converts aperiodic discrete signals into the frequency domain.

The intution for the Discrete Time Fourier Transform is more or less the same as that of the Continuous Time Fourier Transform.

Definition 28

The Inverse Discrete Time Fourier Transform converts the frequency domain representation of a signal back into its time domain representation.

Properties of the DTFT

Frequency Shift:

Time Reversal:

Conjugation:

Time Expansion: In discrete time, compression of a signal doesn’t make sense because we can’t have partial steps (i.e n must be an integer). However, we can stretch a signal.

Derivative Property:

Multiplication Property:

Convolution Property:

Convergence of the DTFT

Just like in continuous time, it was unclear whether or not the integral would converge, in discrete time, it is unclear if the infinite sum will converge. The convergence theorem for both are essentially the same.

Theorem 6

Applying the synthesis equation to this, we get

and we can apply the frequency shift property to get

Definition 29

Discrete Fourier Transform

Whereas the CTFT takes a continuous signal and outputs a continuous frequency spectrum and the DTFT takes a discrete signal and outputs a continuous, periodic frequecy spectrum, the Discrete Fourier Transform takes a discrete periodic signal and outputs a discrete frequency spectrum.

Definition 30

Definition 31

2D Fourier Transforms

So far, our Fourier Transforms have been limited to signals of a single dimension. However, in the real world, signals might be multidimensional (think images). Thankfully, each of the Fourier Transforms generalizes easily into higher dimensions.\

Just like in 1 dimension, absolute summability/integrability guarantee the convergence of these transforms. It turns out that when a 2D signal is simply a multiplication of two 1D signals, the Fourier Transforms are very easy to compute.

Theorem 7

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