Fourier Transform – Chapter 13

Fourier Transform – continuous function Apply the Fourier Series to complex- valued functions using Euler’s notation to get the Fourier Transform And the inverse Fourier Transform

Discrete Signals Next time

Sampling Conversion of a continuous function to a discrete function What does this have to do with the Fourier Transform? –Procedurally – nothing –Analytically – places constraints

Impulse function It all starts with the Dirac (delta) function we looked at previously Area “under” the signal is 1 –Infinitely tall –Infinitesimally narrow –Practically impossible for x ≠ 0

Sampling with the impulse function Multiply the continuous function with the delta function… results in the continuous function at position 0

Sampling with the impulse function Multiply the continuous function with the delta function shifted by x 0 … results in the continuous function at position x 0

Sampling with the impulse function Sampling two points at a time… For N points at a time…

The comb function

Sampling (pointwise multiplication) with Shah (comb function) provides all the sampled points of the original continuous signal at one time The sampling interval can be controlled as follows

The comb function and sampling The Fourier Transform of a comb is a comb function (same situation as we saw with the Gaussian) Combine this with the convolution property of the Fourier Transform The result is that the frequency spectrum of the [original] continuous function is replicated infinitely across the frequency spectrum

The comb function and sampling If the continuous function contains frequencies less than ω max and… the sampling frequency (distance between delta functions of the comb) is at least twice ω max … then all is OK This is referred to as the Nyquist Theorem

The comb function and aliasing If the continuous function contains frequencies less than ω max and… the sampling frequency (distance between delta functions of the comb) is less than twice ω max … then all is you get aliasing This violates the Nyquist Theorem

Aliasing Means the original, continuous signal cannot be uniquely recovered from the sampled signal’s spectrum (Fourier Transform) –Basically, this means there are not enough points in the sampled wave form to accurately represent the continuous signal

Aliasing

Discrete Fourier Transform Now that we know how to properly transform a continuous function to a discrete function (sample) we need a discrete version of the Fourier Transform

Discrete Fourier Transform Forward Inverse M is the length (number of discrete samples)