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.

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

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

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.

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 \(t \in \mathbb{R}\).

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 \(k \in \mathbb{Z}\).