Difference between revisions of "CourseNotes3 (BoutinSpring2009)" - Rhea

Revision as of 13:22, 27 January 2009

ECE438 Course Notes January 21, 2009

Delta Functions

Continuous-time: (a.k.a. Dirac delta function)
$ \delta(t) = \lim_{\triangle \Rightarrow 0} \frac{1}{\triangle}rect(\frac{t}{\triangle}) $
Properties

• $ \int_{-\infty}^{\infty} \delta(t)\,dt = 1 $(unit area)
• $ \int_{-\infty}^{\infty} x(t) \delta(t-t_0)\,dt = x(t_0) $(sifting property)

Discrete-time: (a.k.a. Kronecher delta fn.)
$ \delta[n] = 1 | n=0, 0 | 0 > n < 0 $

• Sifting Property: $ \sum_{n=-\infty}^{\infty} x[n] \delta[n-n_0] = x[n_0] $

Comb & Rep operators (for CT signals)

Comb operator multiplies a signal by an "impulse train".

• $ \sum_{k=-\infty}^{\infty} \delta(t-kT) $
• $ Comb_T{{x(t)}} = x(t)\sum_{k=-\infty}^{\infty} \delta(t-kT) = \sum_{k=-\infty}^{\infty} x(kT)\delta(t-kT) $

Rep operator simply replicates a signal every T units:
$ rep_T{{x(t)}} = \sum_{k=-\infty}^{\infty} x(t-kT) $

Systems

A system maps an input signal x(t) to a unique output signal, y(t). $ x(t) \Rightarrow \mbox {System} \Rightarrow y(t) $
$ y(t) = S[x(t)] $

Examples:
$ y(n) = \frac{1}{3}[x(n) + x(n-1) + x(n-2)] $ (moving averaging function, seen in Lab2)

System Properties:

• Linearity
  • Definition: A system S is linear if for any two input signals $ x_1(t) $ and $ x_2(t) $, and any (complex) constant, a, it satisfies the following two properties:</li?
  • Superposition: $ S[x_1(t) + x_2(t)] = S[x_1(t)] + S[x_2(t)] $
  • Homogeneity: $ S[ax_1(t)] = aS[x_1(t)] $
• Time-Invariance
  • Definition: A system S is time-invariant(TI) if delaying the input signal results only in an identical delayin the output signal.
  • If $ y_1(t) = S[x_1(t)] $ and $ y_2(t) = S[x_1(t-t_0)] $ than $ y_2(t) = y_1(t-t_0) $