## 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:

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 = $\pi$ 