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Signal#

Notation#

Name Description
Time function \(x(t)\)
Spectral function \(X(f)\)
Sampling Time \(T_A = \frac{1}{f_A}\)
Sampling Frequency \(f_A = \frac{1}{T_A}\)
Symbol Notation.

Rectangular Pulse#

mpl{name=rectangle} fig, ax = plt.subplots() N = 100 # sample count P = 10 # period D = 5 # width of pulse sig = np.arange(N) % P < D ax.plot(sig)

Signal Categories#

image/svg+xml t y t y t y t y analog digital time-continuous time-discrete value-discrete value-continues
Signal Categories
Details:

If a signal is both value--continuous and time--continuous, it is called a analog signal. A digital signal, on the other hand, is always value--discrete and time-discrete and the message contained therein consists of the symbols of a symbol set.

  • Time-Continuous Signal: the signal parameter \(t\) is defined at any given time.
  • Time-Discrete Signal: the signal parameter is defined only at the discrete points \(t_v = v \cdot T_A\).
  • Deterministic Signal: the time function \(x(t)\) can be described completely in analytical form.
  • Random Signal: its time function \(x(t)\) is not -- or at least not completely -- describable in mathematical form. Such a signal cannot be predicted exactly for the future.
  • Causal Signal/System: if \(x(t)\) does not exist for all times \(t < 0\) or is identical zero.
  • Energy--Limited Signal: if \(x(t)\) has finite energy \(E_x\) and infinitely small power (\(P_x \to 0\))
  • Power--Limited Signal: if \(x(t)\) has finite power \(P_x\) and infinite energy \((E_x \to \infty)\).
Examples of Signals

Causal vs.Β Non-Causal Signal

You can see a causal system in the upper graphic: If a unit step function \(x(t)\) is applied to its input, then the output signal 𝑦(𝑑) can only increase from zero to its maximum value after time 𝑑=0. In the lower graph the causality is no longer given. As you can easily see in this example, an additional runtime of one millisecond is enough to change from the non-causal to the causal representation.

Energy--Limited and Power--Limited Signals#

Both, energy and power of a signal, are dependent on the resistance 𝑅. In order to eliminate this dependency, the resistance 𝑅=1Ξ© is often used as a basis in communications engineering. Then the following definitions apply:

The energy in \(\si{\text{V}^2\text{s}}\) of the signal \(x(t)\) is:

\[E_x=\lim_{T_{\rm M}\to\infty} \int^{T_{\rm M}/2} _{-T_{\rm M}/2} x^2(t)\,{\rm d}t.\]

with \(T_M\) is the assumed measurement duration during which the signal is observed, symmetrically with respect to the time origin (\(t=0\)). In general, this time interval must be chosen very large; ideally \(T_M\) should be towards infinity.

The (mean) power in \(\si{\text{V}^2}\) is the energy divided by the time before the limit crossing:

\[P_x = \lim_{T_{\rm M} \to \infty} \frac{1}{T_{\rm M} } \cdot \int^{T_{\rm M}/2} _{-T_{\rm M}/2} x^2(t)\,{\rm d}t.\]

with \(T_M\) is the assumed measurement duration during which the signal is observed, symmetrically with respect to the time origin (\(t=0\)). In general, this time interval must be chosen very large; ideally \(T_M\) should be towards infinity.

Example Energy and Power Calculations

Now the energy and power of two exemplary signals are calculated.

Rectangular Pulse

The upper graphic shows a rectangular pulse \(x_1(t)\) with amplitude \(A\) and duration \(T\).

  • The signal energy of this pulse is \(E_1 = A^2 \cdot T\).
  • For the signal power, division by \(T_{\rm M}\) and limit formation \((T_{\rm M} \to \infty)\) the value is \(P_1 = 0\).

For the cosine signal \(x_2(t)\) with amplitude \(A\) applies according to the sketch below:

  • The signal power is \(P_2 = A^2/2\), regardless of the frequency.
  • The signal energy \(E_2\) (integral over power for all times) is infinite.

Signals and Spectrum#

Time \(x(t)\) Frequency \(X(f)\)

References#