Understanding Fourier Transform: Sampling and Aliasing

1. Fourier Transform – Chapter 13

2. 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

3. Discrete Signals Next time

4. 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

5. 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

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

7. Sampling with the impulse function • Multiply the continuous function with the delta function shifted by x0 … results in the continuous function at position x0

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

9. The comb function

10. 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

11. 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

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

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

14. 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

15. Aliasing

16. 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

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