Contents
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) $
