notes/docs/physics/mathematical-physics/signal-analysis/signals.md

Signals

Definitions

Definition: a signal is a function of space and time.

• Output can be analog or quantised.
• Input can be continuous or discrete.

Definition: a signal can be sampled at particular moments k T_s in time, with k \in \mathbb{Z} and T_s \in \mathbb{R} the sampling period. For a signal f: \mathbb{R} \to \mathbb{R} sampled with a sampling period T_s may be denoted by

f[k] = f(kT_s), \qquad \forall k \in \mathbb{Z}.

Definition: signal transformations on a function x: \mathbb{R} \to \mathbb{R} obtaining the function y: \mathbb{R} \to \mathbb{R} are given by

Signal transformation	Time	Amplitude
Reversal	y(t) = x(-t)	y(t) = -x(t)
Scaling	y(t) = x(at)	y(t) = ax(t)
Shifting	y(t) = x(t - b)	y(t) = x(t) + b

for all t \in \mathbb{R}.

For sampled signals similar definitions hold.

Symmetry

Definition: consider a signal f: \mathbb{R} \to \mathbb{R} which is defined in an interval which is symmetric around t = 0, we define.

• f is even if f(t) = f(-t), \forall t \in \mathbb{R}.
• f is odd if f(t) = -f(-t), \forall t \in \mathbb{R}.

For sampled signals similar definitions hold.

Theorem: every signal can be decomposed into symmetric parts.

??? note "Proof:"

Will be added later.

Periodicity

Definition: a signal f: \mathbb{R} \to \mathbb{R} is defined to be periodic in T if and only if

f(t + T) = f(t), \qquad \forall t \in \mathbb{R}.

For sampled signals similar definitions hold.

Theorem: a summation of two periodic signals with periods T_1, T_2 \in \mathbb{R} respectively is periodic if and only if

\frac{T_1}{T_2} \in \mathbb{Q}.

??? note "Proof:"

Will be added later.

Signals

Definition: the Heaviside step signal u: \mathbb{R} \to \mathbb{R} is defined by

u(t) = \begin{cases} 1 &\text{ if } t > 0,\ 0 &\text{ if } t < 0,\end{cases}

for all t \in \mathbb{R}.

For a sampled function the Heaviside step signal is given by

u[k] = \begin{cases} 1 \text{ if } k \geq 0, \ 0 \text{ if } k < 0, \end{cases}

for all k \in \mathbb{Z}.

Definition: the rectangular signal \text{rect}: \mathbb{R} \to \mathbb{R} is defined by

\text{rect} (t) = \begin{cases} 1 &\text{ if } |t| < \frac{1}{2}, \ 0 &\text{ if } |t| > \frac{1}{2},\end{cases}

for all t \in \mathbb{R}.

The rect signal can be normalised obtaining the scaled rectangular signal D: \mathbb{R} \to \mathbb{R} defined by

D(t, \varepsilon) = \begin{cases} \frac{1}{\varepsilon} &\text{ if } |t| < \frac{\varepsilon}{2},\ 0 &\text{ if } |t| > \frac{\varepsilon}{2},\end{cases}

for all t \in \mathbb{R}.

The following signal has been derived from the scaled rectangular signal D: \mathbb{R} \to \mathbb{R} used on a signal f: \mathbb{R} \to \mathbb{R} for

\lim_{\varepsilon ;\downarrow; 0} \int_{-\infty}^{\infty} f(t) D(t, \varepsilon)dt = \lim_{\varepsilon ;\downarrow; 0} \frac{1}{\varepsilon} \int_{-\frac{\varepsilon}{2}}^{\frac{\varepsilon}{2}} f(t) dt = f(0),

using the mean value theorem for integrals.

Definition: the Dirac signal \delta is a generalized signal defined by the properties

\begin{align*} \delta(t - t_0) = 0 \quad \text{ for } t \neq t_0,& \ \int_{-\infty}^\infty f(t) \delta(t - t_0) dt = f(t_0),& \end{align*}

for a signal f: \mathbb{R} \to \mathbb{R} continuous in t_0.

For sampled signals the \delta signal is given by

\delta[k] = \begin{cases} 1 &\text{ if } k = 0, \ 0 &\text{ if } k \neq 0.\end{cases}

Signal sampling

We already established that a signal f: \mathbb{R} \to \mathbb{R} can be sampled with a sampling period T_s \in \mathbb{R} obtaining f[k] = f(kT_s) for all k \in \mathbb{Z}. We can also define a time-continuous signal f_s: \mathbb{R} \to \mathbb{R} that represents the sampled signal using the Dirac signal, obtaining

f_s(t) = f(t) \sum_{k = - \infty}^\infty \delta(t - k T_s), \qquad \forall t \in \mathbb{R}.

Definition: the sampling signal or impulse train \delta_{T_s}: \mathbb{R} \to \mathbb{R} is defined as

\delta_{T_s}(t) = \sum_{k = - \infty}^\infty \delta(t - k T_s)

for all t \in \mathbb{R} with a sampling period T_s \in \mathbb{R}.

Then integration works out since we have

\int_{-\infty}^\infty f(t) \delta_{T_s}(t) dt = \sum_{k = -\infty}^\infty \int_{-\infty}^\infty f(t) \delta(t - k T_s) dt = \sum_{k = -\infty}^\infty f [k],

by definition.

Convolutions

Definition: let f,g: \mathbb{R} \to \mathbb{R} be two continuous signals, the convolution product is defined as

f(t) * g(t) = \int_{-\infty}^\infty f(u)g(t-u)du

for all t \in \mathbb{R}.

Proposition: the convolution product is commutative, distributive and associative.

??? note "Proof:"

Will be added later.

Theorem: let f: \mathbb{R} \to \mathbb{R} be a signal then we have for the convolution product between f and the Dirac signal \delta and some t_0 \in \mathbb{R}

f(t) * \delta(t - t_0) = f(t - t_0)

for all t \in \mathbb{R}.

??? note "Proof:"

let $f: \mathbb{R} \to \mathbb{R}$ be a signal and $t_0 \in \mathbb{R}$, using the definition of the Dirac signal 

$$
    f(t) * \delta(t - t_0) = \int_{-\infty}^\infty f(u) \delta(t - t_0 - u)du = f(t - t_0),
$$

for all $t \in \mathbb{R}$. 

In particular f(t) * \delta(t) = f(t) for all t \in \mathbb{R}; \delta is the unity of the convolution.

The average value of a signal f: \mathbb{R} \to \mathbb{R} for an interval \varepsilon \in \mathbb{R} may be given by

f(t) * D(t, \varepsilon) = \frac{1}{\varepsilon} \int_{t - \frac{\varepsilon}{2}}^{t + \frac{\varepsilon}{2}} f(u)du.

For sampled/discrete signals we have a similar definition for the convolution product, given by

f[k] * g[k] = \sum_{m = -\infty}^{\infty} f[m]g[k - m],

for all k \in \mathbb{Z}.

Correlations

Definition: let f,g: \mathbb{R} \to \mathbb{R} be two continuous signals, the cross-correlation is defined as

f(t) \star g(t) = \int_{-\infty}^\infty f(u) g(t + u)du

for all t \in \mathbb{R}.

Especially the auto-correlation of a continuous signal f: \mathbb{R} \to \mathbb{R} given by f(t) \star f(t) for all t \in \mathbb{R} is useful, as it can detect periodicity. This is proved in the section Fourier series.

For sampled/discrete signals a similar definition exists given by

f[k] \star g[k] = \sum_{m = -\infty}^{\infty} f[m]g[k + m],

for all k \in \mathbb{Z}.