MIT 6.007 Signals and Systems Course Notes 1

Visit original course website for more details.

Lecture 1: Introduction

Signals

A Simple Definition

Functions of one or more independent variables.

Types of Signals

• Continuous-time signals
• Discrete-time signals
• One-dimensional signals
• Multi-dimensional signals

Examples:

Speech signal: continuous-time, one-dimensional

Image signal: continuous-time, two-dimensional

stock market index: discrete-time, one-dimensional

Systems

A Simple Definition

How we process signals. Typically has an input and an output.

Types of Systems

• Linear/Non-linear systems
• Time-invariant/Time-varying systems

Focus more on Linear Time-invariant(LTI) systems.

Domains

Time domain and frequency domain.

Lecture 2: Signals and Systems Part I

Sinusoidal signal

Continuous-time

\[x(t) = A\cos(\omega_0 t + \phi)\]

Properties

• Periodic: \(T = \frac{2\pi}{\omega_0}\)

• Time-shift <=> Phase change

  A time shift of \(\Delta t\) can generate a phase change of \(\Delta \phi\) and vice versa satisfying the following equation:

  \[\omega_0\Delta t = \Delta \phi\]
• Odd(\(\sin\)) and Even(\(\cos\))
• Different signals for different choice of \(\omega_0\)

Discrete-time

\[x[n] = A\cos(\Omega_0 n + \phi)\]

Due to the constraint that \(n \in \mathbb{Z}\)，some properties in continuous-time case do NOT hold.

Properties

• Periodic

  Holds only if when \(\Omega_0 = \frac{2\pi m}{N}, m \in \mathbb{Z}, N \in \mathbb{N}^{*}\)

• Time-shift => Phase-change

  A time shift of \(\Delta n\) can generate a phase change of \(\Delta \phi\) while the opposite does not always hold since \(\frac{\Delta \phi}{\Omega_0}\) can not be guaranteed to be an integer.

• Odd(\(\sin\)) and Even(\(\cos\))
• Identical signals for values of \(\Omega_0\) separated by \(2\pi\).

Real Exponential

Continuous-time

\[x(t) = C\text{e}^{at}, C,a \in \mathbb{R}\]

Time-shift <=> Scale change

Discrete-time

\[x[n] = C\alpha^n, C,\alpha \in \mathbb{R}\]

Note that \(\alpha\) can be negative here.

Complex Exponential

Continuous-time

\[x(t) = C\text{e}^{a t}, C,a \in \mathbb{C}\]

Write \(C\) in the polar form and \(a\) in the rectangular coordinate form:

\[C=|C|\text{e}^{j\theta}\] \[a=r+j\omega_0\]

We get:

\[\begin{align} x(t) &= |C|\text{e}^{(r+j\omega_0)t+j\theta} = |C|\text{e}^{rt}\text{e}^{j(\omega_0 t+\theta)} \notag \\ &= |C|\text{e}^{rt}\cos(\omega_0 t + \theta) + j|C|\text{e}^{rt}\sin(\omega_0 t + \theta) \notag \end{align}\]

Discrete-time

\[x[n] = C\alpha^{n}, C,\alpha \in \mathbb{C}\]

Write \(C, \alpha\) in the polar form:

\[C=|C|\text{e}^{j\theta}\] \[\alpha=|\alpha|\text{e}^{j\Omega_0}\]

We get:

\[\begin{align} x[n] &= |C|\text{e}^{j\theta}(|\alpha|\text{e}^{j\Omega_0})^n = |C||\alpha|^n \text{e}^{j(\Omega_0 n+\theta)} \notag \\ &= |C||\alpha|^n \cos(\Omega_0 n + \theta) + j|C||\alpha|^n\sin(\Omega_0 n + \theta) \notag \end{align}\]

Lecture 3: Signals and Systems Part II

Unit Step and Unit Impulse

Discrete Form

\[\begin{align} u[n] &= \begin{cases} 1 & n \geq 0 \\ 0 & n < 0 \end{cases} \notag \\ \delta[n] &= \begin{cases} 1 & n=0 \\ 0 & n \neq 0 \end{cases} \notag \end{align}\]

Unit Impulse can be regarded as taking the first difference of unit step, i.e. \(\delta[n] = u[n] - u[n-1]\), thus we can reconstruct the unit step from the unit impulse by taking the summation of the unit impulse: \(u[n]=\sum_{k=0}^{\infty}\delta[n-k]= \sum_{m=-\infty}^n \delta[m]\).

Continuous Form

\[\begin{align} u(t) &= \begin{cases} 1 & t > 0 \\ 0 & t < 0 \end{cases} \notag \end{align}\]

How to deal with \(t=0\) case?

Use an approximation:

\[\begin{align} u_{\Delta}(t) &= \begin{cases} 0 & t < 0 \\ \frac{t}{\Delta} & 0 \leq t < \Delta \\ 1 & t \geq \Delta \end{cases} \notag \end{align}\]

Thus we can define \(u(t)\) as the limit of its approximation when \(\Delta\) goes to zero.

Also we can define unit impulse in the continuous form by first taking derivative of \(u_{\Delta}(t)\) and then let \(\Delta\) goes to zero. The \(u(t)\) defined below satisfies that the area around zero (\(0*\infty\)) equals 1.

\[\begin{align} \delta(t) &= \begin{cases} \infty & t=0 \\ 0 & t \neq 0 \end{cases} \notag \end{align}\]

We can reconstruct unit step function by taking the integral of unit impulse:

\[\delta(t) = \frac{\text{d}}{\text{d}t}u(t)\] \[u(t) = \int_{-\infty}^t \delta(\tau) \text{d}\tau\]

Systems, Interconnections of systems and Properties of systems

Systems

A system is a mapping from an input signal to an output signal. The mapping is usually denoted by a box with an arrow pointing to the input and another arrow pointing to the output.

Interconnections of systems

We can connect systems in series or in parallel.

Series

The output of one system is the input of the other.

Parallel

The output of the parallel system is the sum of the outputs of the two systems.

Feedback

The output of the system 1 is fed back by system 2 to the input.

Properties of systems

Memoryless

A system is memoryless if the output at time \(t\) depends only on the input at time \(t\).

Example: Squarer(yes), Unit delay(no).

Invertibility

A system is invertible if there exists another system that can undo the effect of the first system (to make up an identity system together).

Example: Differentiator is the inverse system of integrator, thus integrator is invertible. However the opposite does NOT hold since there exists an additional const \(C\) when you do the integration after differentiation.

Causality

A system is causal if the output at time \(t\) depends only on the input at time \(t\) and before.

Example: \(y[n] = x[n] + x[n-1]\) is causal, While \(y[n] = \frac{1}{3}(x[n-1]+x[n]+x[n+1])\) isn’t.

Stability

A system is stable if the output is bounded for all bounded inputs.

Example: \(y(t) = \int_{-\infty}^tx(\tau)\text{d}\tau\) isn’t.

Time Invariance

\(x(t)\to y(t)\), then \(x(t-t_0)\to y(t-t_0)\).

Linearity

\(x_1(t)\to y_1(t)\), \(x_2(t)\to y_2(t)\), then \(ax_1(t)+bx_2(t)\to ay_1(t)+by_2(t)\).