Introduction Earlier you should have been exposed to the concepts behind sampling and the sampling theorem. While learning about these ideas, you should have begun to notice that if we sample at too low of a rate, there is a chance that our original signal will not be uniquely defined by our sampled signal. If this happens, then there is no guarantee that we can correctly reconstruct the signal. As a result of this, the Nyquist Theorem was created. Below, we will discuss just what exactly this theorem tells us.

Nyquist Theorem We will let T equal our sampling period (distance between samples). Then let Ωs 2 T (sampling frequency in radians/sec). We have seen that if f t is bandlimited to ΩB ΩB and we sample with period T Ωb 2 Ωs ΩB Ωs 2 ΩB then we can reconstruct f t from its samples. Nyquist Theorem ("Fundamental Theorem of DSP") If f t is bandlimited to ΩB ΩB , we can reconstruct it perfectly from its samples fs n f n T for Ωs 2 T 2 ΩB ΩN 2 ΩB is called the "Nyquist frequency" for f t . For perfect reconstruction to be possible Ωs 2 ΩB where Ωs is the sampling frequency and ΩB is the highest frequency in the signal.

Illustration of Nyquist Frequency Examples: Human ear hears frequencies up to 20 kHz → CD sample rate is 44.1 kHz. Phone line passes frequencies up to 4 kHz → phone company samples at 8 kHz.

Reconstruction The reconstruction formula in the time domain looks like f t n fs n T t n T T t n T We can conclude, just as before, that n n T t n T T t n T is a basis for the space of ΩB ΩB bandlimited functions, ΩB T . The expansion coefficient for this basis are calculated by sampling f t at rate 2 T 2 ΩB . The basis is also orthogonal. To make it orthonormal, we need a normalization factor of T .

The Big Question What if Ωs 2 ΩB ? What happens when we sample below the Nyquist rate? Go through the steps: (see )

Finally, what will happen to F s ω now? To answer this final question, we will now need to look into the concept of aliasing.