Linearity

Linear system: y[n] = 8x[n/4]

Non-linear system: y(t) = 5x^2(t) + 11

Causality

Causal system: y[n] = x[n] + x[n-4]

Non-causal system: y(t) = 3x(-t/6)

Memory

System w/ memory: y[n] = 3t^2x[n-9]

Memoryless system: y(t) = (4x(t))^2

Invertibility

Invertible system: y[n] = 2x[n+4]

Non-invertible system: y(t) = x^4(t) -5

Stability

Stable system: y[n] = x^2[n] +4x[n] -1

Non-stable system: y(t) = d^2x(t)/dt^2 + dx(t)/dt + 6x(t)

Time Invariance

Time-variant system: y[n] = 3n^2x[n] + 5

Time-invariant system: y(t) = 2x(t) + x(t-3)

Question

Determine and sketch the convolution of the following signals:

x(t) = t+1 for $ 0 \le t \le 1 $

x(t) = 2-t for $ 1 < t \le 2 $

x(t) = 0 everywhere else

h(t) = $ \delta(t+2) $ + $ 2\delta(t+1) $

Solution

x(t)*h(t) = $ \int x(\tau)h(t-\tau)d\tau $ = $ \int h(\tau)x(t-\tau)d\tau $

Becomes x(t)*h(t) = x(t+2)+2x(t+1)

Sketches:

Bonus1.jpg

y(t) = t+3 for $ -2 < t \le -1 $

y(t) = t+4 for $ -1 < t \le -0 $

y(t) = 2-2t for $ 0 < t \le 1 $

y(t) = 0 everywhere else

Question

Determine the fundamental period of the signal x(t) = 5cos(14t)+sin(6t-2).

Solution

Period of first term = $ \frac{2\pi}{14} $ = $ \frac{\pi}{7} $

Period of second term = $ \frac{2\pi}{6} $ = $ \frac{\pi}{3} $

Least common multiple = Fundamental period = $ \pi $



Please place solutions and/or comments below.

Back to first bonus point opportunity, ECE301 Spring 2013

Alumni Liaison

Questions/answers with a recent ECE grad

Ryne Rayburn