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