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Signals and Systems
Lecture 3
DR TANIA STATHAKI READER (ASSOCIATE PROFESSOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE LONDON
Number of samples kept symmetrically around the origin.
DFT Properties
𝑲 − 𝑹 − −𝑲 + 𝟏 = 𝟐𝑲 + 𝟏 − 𝑹 = 𝑴𝑵 terms
𝑲 − 𝑲 − 𝑹 + 𝟏 + 𝟏 =𝑹 terms
Proof of Case 2
Symmetries
Hermitian: A complex matrix that is equal to its own conjugate transpose.
𝑮𝑯𝑮 =𝟏
𝑵𝑭𝑯
𝟏
𝑵𝑭 =
𝟏
𝑵𝑭𝑯𝑭
=𝟏
𝑵𝑵𝑭−𝟏𝑭 = 𝑰
Parseval’s Theorem
−𝟏 𝟏 𝟏 𝟐 𝟎 − 𝟏 −𝟐 𝟐 𝟑 − 𝟏 − 𝟏 −𝟑 𝟏 𝟑
−𝟐 𝟐 𝟑 − 𝟏 − 𝟏
Convolution
∗
eliminated
After convolution with a lowpass filter the signal
becomes smoother.
× After sampling CTFT (same as DTFT)
becomes periodic.
Lowpass filter the signal in order to make it bandlimited for sampling. Window the signal to make it of finite duration.
Sampling Process
Zero Padding
Phase Unwrapping
Uncertainty Principle
Uncertainty Principle Proof Steps
Summary
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