System Properties Especially: Linear Time Invariant Systems

Scaling Homogeneity: Scaling the input is the same as scaling the output signal. CT: ℋ 𝑘𝑥 𝑡 =𝑘ℋ 𝑥 𝑡 DT: ℋ 𝑘𝑥 𝑛 =𝑘ℋ 𝑥 𝑛

Adding Additivity: Adding input signals is the same as adding individual output signals. CT: ℋ 𝑖=1 𝑀 𝑥 𝑖 𝑡 = 𝑖=1 𝑀 ℋ 𝑥 𝑖 𝑡 DT: ℋ 𝑖=1 𝑀 𝑥 𝑖 𝑛 = 𝑖=1 𝑀 ℋ 𝑥 𝑖 𝑛

Linear: Scaling and Adding Linearity: Having both Homogeneity and Additivity. Only need to show for 𝑀= 2 since induction can then be used to show it holds for all positive 𝑀. CT: ℋ 𝑖=1 𝑀 𝑘 𝑖 𝑥 𝑖 𝑡 = 𝑖=1 𝑀 𝑘 𝑖 ℋ 𝑥 𝑖 𝑡 DT: ℋ 𝑖=1 𝑀 𝑘 𝑖 𝑥 𝑖 𝑛 = 𝑖=1 𝑀 𝑘 𝑖 ℋ 𝑥 𝑖 𝑛

Time Invariance Time shifting the input is the same as time shifting the output signal. CT: ℋ 𝑥 𝑡−𝜏 = ℋ 𝑥 𝜆 𝜆=𝑡−𝜏 DT: ℋ 𝑥 𝑛−𝑁 = ℋ 𝑥 𝑝 𝑝=𝑛−𝑁 On the left H is acting on x as a function of t. Since we are using a substitution of variables on the right, H is acting on x as a function of t. This distinction will become more obvious in the examples.

LTI Having both Linearity and Time Invariance. CT: ℋ 𝑖=1 𝑀 𝑘 𝑖 𝑥 𝑖 𝑡− 𝜏 𝑖 = 𝑖=1 𝑀 𝑘 𝑖 ℋ 𝑥 𝑖 𝜆 𝜆=𝑡− 𝜏 𝑖 DT: ℋ 𝑖=1 𝑀 𝑘 𝑖 𝑥 𝑖 𝑛− 𝑁 𝑖 = 𝑖=1 𝑀 𝑘 𝑖 ℋ 𝑥 𝑖 𝑝 𝑝=𝑛− 𝑁 𝑖 Fundament requirement for system analysis discussed in following chapters.

Other Properties Bounded-Input Bounded-Output Stability Causality If the input 𝑥 𝑛 ∞ <∞, then the output 𝐻 𝑛 ∞ <∞. ⋅ ∞ means the maximum absolute value Causality The output does not occur before the input. Said another way: the output at time 𝑡 does not depend on future input values (e.g. 𝑡+1). Invertibility There is a unique output for every unique input. This means that if you know the output, then you can figure out what the input is. E.g. 𝑦=2𝑥 is invertible, but 𝑦= 𝑥 2 is not. Static vs. Dynamic Static: the output depends on the input at the same time. Dynamic: the output depends on input values at other times.

Example: H[x(t)] = 3x(t) + 1 Linearity: ℋ 𝑘 1 𝑥 1 𝑡 +𝑘 2 𝑥 2 𝑡 = 𝑘 1 ℋ 𝑥 1 𝑡 + 𝑘 2 ℋ 𝑥 2 𝑡 3k1x1(t) + 1 + 3k2x2(t) + 1 = k1(3x1(t) + 1) + k2(3x2(t) + 1)? 3k1x1(t) + 1 + 3k2x2(t) + 1 = k13x1(t) + k1 + k23x2(t) + k2? NO does 2 = k1 + k2? This equality does not hold for all values of k1,2. Time Inv.: ℋ 𝑥 𝑡−𝜏 = ℋ 𝑥 𝜆 𝜆=𝑡−𝜏 3x(t – τ) + 1 = (3x(λ) + 1)|λ=t – τ? YES 3x(t – τ) + 1 = 3x(t – τ) + 1? This holds for all values of τ.

Example: y(t) = H[x(t)] = 3x(t) + 1 BIBO Yes, If the input is bounded the output is bounded to 3 times the input plus 1. Causality Yes, does not depend on future values. Invertible Yes, x(t) = (1/3)(y(t) – 1). Dynamic or Static Static, only depends on the input at the same time.

Example: H[x(t)] = 5x(t+2) Linearity: ℋ 𝑘 1 𝑥 1 𝑡 +𝑘 2 𝑥 2 𝑡 = 𝑘 1 ℋ 𝑥 1 𝑡 + 𝑘 2 ℋ 𝑥 2 𝑡 5k1x1(t+2) + 5k2x2(t+2) = k15x1(t+2) + k25x2(t+2)? YES This equality holds for all values of k1,2. Time Inv.: ℋ 𝑥 𝑡−𝜏 = ℋ 𝑥 𝜆 𝜆=𝑡−𝜏 5x(t+2–τ) = (5x(λ+2))|λ=t–τ? YES 5x(t+2–τ) = 5x(t–τ+2)? This holds for all value of τ.

Example: y(t) = H[x(t)] = 5x(t+2) BIBO Yes, If the input is bounded the output is bounded to 5 times the input. Causality No, it does depend on future values. Invertible Yes, x(t) = (1/5)y(t-2). Dynamic or Static Dynamic, depends on the input at different time(s).

Example: H[x[n]] = nx[n] Linearity: ℋ 𝑘 1 𝑥 1 𝑛 +𝑘 2 𝑥 2 𝑛 = 𝑘 1 ℋ 𝑥 1 𝑛 + 𝑘 2 ℋ 𝑥 2 𝑛 nk1x1[n] + nk2x2[n] = k1nx1[n] + k2nx2[n]? YES This equality holds for all values of k1,2. Time Inv.: ℋ 𝑥 𝑛−𝑁 = ℋ 𝑥 𝑝 𝑝=𝑛−𝑁 nx[n–N] = px[p]|p=n–N? NO nx[n–N] = (n-N)x[n–N]? This does not hold for all value of N.

Example: H[x[n]] = nx[n] BIBO No, the 𝐻 𝑥 𝑛 →∞ as 𝑛→∞ for bounded nonzero 𝑥 𝑛 Causality No, it does depend on future values. Invertible No, x[n] = (1/n)y[n] works except for n=0. At n=0, the output will be zero regardless of the input. Can’t determine x at n=0. Dynamic or Static Static, only depends on the input at the same time.

Example: H[x(t)] = sin(x(t)) Linearity: ℋ 𝑘 1 𝑥 1 𝑡 +𝑘 2 𝑥 2 𝑡 = 𝑘 1 ℋ 𝑥 1 𝑡 + 𝑘 2 ℋ 𝑥 2 𝑡 No why? Time Inv.: ℋ 𝑥 𝑡−𝜏 = ℋ 𝑥 𝜆 𝜆=𝑡−𝜏 Yes why?

Example: y(t) = H[x(t)] = sin(x(t)) BIBO Yes, why? Causality Yes, why? Invertible No, why? Dynamic or Static Static, why? How do any of these change with y(t) = tan(x(t))

Common LTI system operators Summation of two signals (additivity): as long as x1 and x2 are both inputs or they are produced through LTI combinations of the same signal.

Common LTI system operators Scaling (homogeneity) by a constant

Common LTI system operators Time Shift (Time Invariance): predominantly used in DT systems. Figure shows delays, but can also have lead or negative delay, 𝑧 𝑁

Common LTI system operators Differentiation: CT systems only.

Common LTI system operators Integration: CT systems only.

Linear Combinations of LTI operators are also LTI. Differences and Accumulation are two examples for DT LTI systems.

Why is LTI so important How can we use the properties of LTI to find y[n] = H[x[n]]? given we know h[n]= H[δ[n]] (impulse response). Any DT signal can be expressed as a sum (additivity) of time-shifted (time invariance) and scaled (homogeneity) unit impulses: 𝑥 𝑛 = 𝑖=−∞ ∞ 𝑘 𝑖 𝛿 𝑛− 𝑁 𝑖 = 𝑖=−∞ ∞ 𝑥 𝑁 𝑖 𝛿 𝑛− 𝑁 𝑖 The output is in a form similar to the LTI equation on the previous slide. 𝑦 𝑛 =ℋ 𝑥 𝑛 =ℋ 𝑖=−∞ ∞ 𝑥 𝑁 𝑖 𝛿 𝑛− 𝑁 𝑖 And using the property of LTI, the output can be expressed in terms of h[n]. 𝑦 𝑛 = 𝑖=−∞ ∞ 𝑥 𝑁 𝑖 ℋ 𝑥 𝑖 𝑝 𝑝=𝑛− 𝑁 𝑖 = 𝑖=−∞ ∞ 𝑥 𝑁 𝑖 ℎ 𝑛− 𝑁 𝑖 More on this in the next chapter … Chapter 5.