# 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#

## 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

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 π¦(π‘) can only increase from zero to its maximum value after time π‘=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 π . In order to eliminate this dependency, the resistance π =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:

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:

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.

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 \(X(f)\) |
---|---|