Signal and System I The unit step response of an LTI system

Signal and System I Linear constant-coefficient difference equations When n  1, Causality + delay

Signal and System I Linear constant-coefficient difference equations + delay Determine A by initial condition: When n = 0, A = 1

Signal and System I Linear constant-coefficient difference equations Two ways: (1) Repeat the procedure (2) + delay

Signal and System I Linear constant-coefficient difference equations When t>0, Determine A by initial condition: Causality +

Signal and System I Linear constant-coefficient difference equations Determine A by initial condition: A = 1 +

Signal and System I Linear constant-coefficient difference equations +

Signal and System I Fourier series representation of continuous-time periodical signal for all tPeriodic signal k is an integer form a complete and orthogonal bases Complete: no other basis is needed. Fourier series Orthogonal: Kronecker Delta

Signal and System I Fourier series representation of continuous-time periodical signal for all tPeriodic signal k is an integer

Signal and System I Fourier series representation of continuous-time periodical signal for all tPeriodic signal k is an integer e.g.

Signal and System I Fourier series representation of continuous-time periodical signal 0

Signal and System I The response of system to complex exponentials Band limited channel Bandwidth

Signal and System I Fourier series representation of discrete-time periodical signal for all t Periodic signal

Signal and System I Example #1

Signal and System I Properties of discrete-time Fourier series (1) Linearity

Signal and System I (2) Time shifting (3) Time reversal

Signal and System I (4) Time scaling (5) multiplication

Signal and System I (6) Conjugation and conjugate symmetry Real signal Even Real & Even

Signal and System I (7) Parseval’s relation

Signal and System I (8) Time difference (9) Running sum

Signal and System I Example N = 4 [1, 2, 2, 1] [1, 1, 1, 1]

Signal and System I Fourier series and LTI system Periodic signal System response doesn’t have to be periodic. Output periodic?

Signal and System I

Filtering Frequency-shaping filters Frequency-selective filters (1) Frequency-shaping filters

Signal and System I (1) Frequency-shaping filters

Signal and System I (2) Frequency-selective filters Low-pass high-pass band-pass

Signal and System I Discrete-time

Signal and System I Example: averaging

Signal and System I Continuous-time Fourier transform Aperiodic signal k is an integer Periodic signal

Signal and System I Continuous-time Fourier transform Aperiodic signal k is an integer Periodic signal

Signal and System I Examples

Signal and System I Properties of continuous-time Fourier transform (1) Linearity

Signal and System I Properties of continuous-time Fourier transform (2) Time shifting (3) Time reversal

Signal and System I Properties of continuous-time Fourier transform (4) Time scaling

Signal and System I Properties of continuous-time Fourier transform (5) Conjugation and conjugate summary Real

Signal and System I Example even Even and real

Signal and System I Differential

Signal and System I Integral

Signal and System I Example

Signal and System I Example

Signal and System I Example

Signal and System I Example

Signal and System I Parseval’s relation

Signal and System I Parseval’s relation for continuous-time Fourier series Parseval’s relation for continuous-time Fourier transfer

Signal and System I Example

Signal and System I Example

Signal and System I Example, P (1) real (2) (3) Solution:

Signal and System I Example, P (1) real (2) (3) Solution:

Signal and System I Example, P Solution:

Signal and System I Example, P (3) Solution:

Signal and System I Example, P (3) Solution: (1) real

Signal and System I Multiplication

Signal and System I Example #1

Signal and System I Example #2

Signal and System I Frequency-selective filtering with variable center frequency x Low pass filter x 1