Unformatted text preview: EE 338 Discrete‐time Signals and Systems Part 2: The Sampling Theorem 1. Ideal impulse‐train sampling: • Ideal periodic sampling: given a continuous‐time signal . discrete‐time representation is o Sampling period: , sampling frequency in Hz: Ω • 2 ∑ Ω ∑ Ω Ω : Copies of Ω are and the sampling period , the , sampling frequency in rad/sec: Impulse train modulation: ∑ o The impulse train: o Impulse train modulation: Time domain Frequency domain • shifted by Ω , and superimposed to generate Ω . Ω is periodic with period Ω . Nyquist sampling theorem 0 for all |Ω| Ω . with Ω o A bandlimited signal o If Ω can be perfectly reconstructed from its sampled version 2Ω , then . (Neighboring copies of Ω do not overlap.) 2Ω , adjacent copies of Ω overlap and we observe aliasing distortion. o If Ω o Nyquist rate: the frequency 2Ω that must be exceeded by the sampling frequency 2. Reconstruction of a continuous‐time signal • In the frequency domain: cutoff frequency : • Time domain by a factor of 3. Anti‐aliasing filtering • Before sampling, let the continuous‐time signal aliasing filter) o o Ω with cutoff frequency :
Ω Ω Ω , |Ω| 0, Ω
Ω Ω , where Ω is an ideal low pass filter with Ω ∑ : copies of are shifted by . , scaled , and superimposed to produce the reconstructed signal pass an ideal low pass filter (the anti‐ Ω 1, |Ω|
Ω 0, Force the sampled signal to be bandlimited to frequencies below Ω and avoid aliasing. Limit the additive noise spectrum and other interference. ...
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- Spring '10
- Frequency, Ω, ideal low pass, Impulse train modulation, Ideal periodic sampling, continuous‐time signal