EECS20N: Signals and Systems

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Week 13

• Sampling
• Sinusoids
• Periodicity
• Aliasing
• Images
• Moire patterns
• Fonts
• Avoiding aliasing
• Reconstruction
• A Model
• Sampling theorem
• Quantization

Sampling

Consider a continuous-time signal x : Reals → Complex. Computers, and digital hardware in general, have difficulty dealing with such signals. An integral part of what it means to be digital is that actions occur in discrete steps, not as a continuous evolution. Computers, however, can manipulate a closely related signal that is constructed by sampling the continuous-time signal.

A sampler is a system

Sampler_T : [Reals → Complex] → [Integers → Complex]

where T is the sampling period (it has units of seconds/sample). The sampling frequency is f_s = 1/T, in units of samples/second (or sometimes Hz, cycles/second). If y = Sampler_T (x) then for all n in Integers,

y(n) = x(nT).

Exercise: Verify that Sampler_T is linear but not time invariant.