signals.md

Signals

Analog signals

Sine waves

Superposition of two sine waves:

$$ \begin{array}{rcl} S & = & \displaystyle A . \sin(\frac{2 \pi}P + \alpha) \\\ \sf position & = & \sf amplitude \times \sin(\frac{2 \pi}{period} + phase) \end{array} $$

$$ \begin{array}{rcl} F & = & 2S \\\ \sf superposition & = & \sf 2 \times position \end{array} $$

Wavelength

The wavelength is a distance a signal can travel in one period.

$$ \begin{array}{rcl} \lambda & = & \displaystyle \frac{v}{f} \\\ \sf wavelength & = & \sf \frac{propagation\ speed}{frequency} \end{array} $$

Example

Wavelength of red light where $f=4.10^{14}$ Hz.

$$ \begin{array}{rcl} \lambda & = & \frac{v}{f} \\\ & = & \frac{3.10^8}{4.10^{14}} \\\ & = & \bf 0,75 \sf \ mm \end{array} $$

Fourier analysis

Jean-Baptiste Fourier showed that any composite signal is actually a combination of simple sine waves with different frequencies, amplitudes, and phases.

The frequency $f$ of the sine wave which is the same as the frequency of the composite signal is called the fundamental frequency or first harmonic.

Digital signals

A digital signal is a composite signal with frequencies between $0$ and $\infty$. We can transmit a digital signal by using baseband or modulation.

If a signal has $L$ levels, each level needs $\log_2{L}$ bits.

$$ \begin{array}{rcl} L & = & B^2 \\\ \sf signal\ length & = & \sf bits^2 \end{array} $$

Bit rate

The bit rate is the number of bits sent in 1 s expressed in bits per second (bps).

$$ \begin{array}{rcl} B & = & F . R . P \\\ \sf bits & = & \sf frames \times resolution \times bits\ per\ pixel \end{array} $$

Example

We need to download text documents at the rate of 100 pages per minute. What is the required bit rate of the channel?

• 24 lines with 80 characters
• one character requires 8 bits

$$ \begin{array}{rcl} 100 . 24 . 80 . 8 & = & \frac{1,536,000}{60} \\\ & = & \bf 25,600 \sf \ bps \end{array} $$

Fourier analysis (digital)

The Fourier analysis can be used to decompose a digital signal.

Type	Description
Periodic	The decomposed signal has a frequency domain representation with an infinite bandwidth and discrete frequencies.
Non-periodic	The decomposed signal still has an infinite bandwidth but the frequencies are continuous.

Baseband transmission

Baseband requires a low-pass channel, a channel with bandwidth that starts from $0$.

If we want to preserve the exact form of a non-periodic digital signal, we need to send the continuous range of frequencies beetween $0$ and $\infty$.

In a low-pass channel with limited bandwidth, we approximate the digital signal with an analog signal.

Modulation tranmission

Modulation allows us to use a bandpass channel — A channel with a bandwidth that does not start from $0$.

Amplification

$$ \begin{array}{rcl} P_2 & = & P_1 . 10^{G / 10} \\\ P_2 & = & P_1 . \sf 10^{gain / 10} \end{array} $$

Transmission impairment

Signal travels through transmission media which are not perfect. The imperfection causes signal impairment. What is sent is not what is received. Three causes of impairment:

Attenuation

Attenuation means loss of energy. When a signal travels through a medium, it loses some of its energy to overcome the resistance of the medium.

To show that a signal has lost or gained strength we use the UMT of the decibel. The decibel (dB) measures the relative strengths of the two signals or one signal at two different points.

$$ \begin{array}{rcl} dB & = & L . D \\\ \sf total\ loss & = & \sf loss\ per\ kilometer \times distance \end{array} $$

$$ dB = 10 . \log_{10}\frac{P_2}{P_1} $$

Where $P_1$ and $P_2$ are the powers of a signal at points $1$ and $2$ respectively.

Example

$$ \begin{array}{rcl} \log{a}.b & = & \log{a} + \log{b} \\\ 10 \log_{10}\frac{P_3}{P_1} & = & 10 \log_{10}\frac{P_3}{P_2} . \frac{P_2}{P_1} \\\ & = & 10 (\log\frac{P_3}{P_2} + \log\frac{P_2}{P_1}) \\\ & = & 10 \log\frac{P_3}{P_2} + 10 \log\frac{P_2}{P_1} \\\ & = & 10 - 3 \\\ & = & \bf 7 \sf \ dB \end{array} $$

Distortion

Distortion means that the signal changes its form or shape because signal components at the receiver have phases different from what they had at the sender.

Noise

Type	Description
Thermal noise	Random motion of electrons in a wire which creates an extra signal not originally sent by the transmitter.
Induced noise	Comes from sources such as motors & appliances.
Crosstalk	An effect of one wire on the other.
Impulse noise	A signal with high energy in a very short time that comes from power lines or lightning.

The Signal-to-Noise Ratio (SNR) is actually the ratio of what is wanted (signal) to what is not wanted ().

$$ \begin{array}{rcl} SNR & = & \displaystyle \frac{ASP}{ANP} \\\ \sf signal-to-noise\ ratio & = & \sf \frac{average\ signal\ power}{average\ noise\ power} \end{array} $$

$SNR_{dB}$ is defined:

$$ \begin{array}{rcl} SNR_{dB} & = & 10 \log_{10} SNR \\ \sf (signal-to-noise\ ratio){dB} & = & 10 \log{10} signal-to-noise\ ratio \end{array} $$

Data rate limits

Noiseless channel

For a noiseless channel, the Nyquist bit rate formula defines the theoretical maximum bit rate.

$$ \begin{array}{rcl} R & = & 2 . B . \log_2 L \\\ \sf bit\ rate (in\ bps) & = & \sf 2 \times bandwidth \times \log_2 signal\ levels\ used \end{array} $$

Noisy channel

In 1944, Claude Shannon introduced a formula called the Shannon capacity to determine the theoretical highest data rate for a noisy channel:

$$ \begin{array}{rcl} C & = & B . \log_2{(1 + SNR)} \\\ \sf capacity (in\ bps) & = & \sf bandwidth \times \log_2{(1 + signal-to-noise\ ratio)} \end{array} $$

For practical purposes when the SNR is very high, we can assume that $SNR+1$ is almost the same as SNR.

Performance

Performance of network — How good is it?

An increase in bandwidth in Hz means an increase in bandwidth in bps.

Measurement	Description
Bandwidth	Potential measure of a link data rate.
Throughput	Actual measure of how fast we can send data.

Latency

The latency or delay defines how long it takes for an entire message to completely arrive at the destination from the time the first bit is sent from the source.

$$ \begin{array}{rcl} L & = & PT + TT + QT + PD \\\ \sf latency & = & \sf propagation\ time + transmission\ time + queueing\ time + processing\ delay \end{array} $$

Propagation time

Propagation time measures time required to travel from the source to the destination.

$$ \begin{array}{rcl} PT & = & \displaystyle \frac{D}{PS} \\\ \sf propagation\ time & = & \sf \frac{distance}{propagation\ speed} \end{array} $$

The propagation speed of electromagnetic signals depends on the medium and frequency of the signal.

Transmission time

The time required for transmission of a message depends on the size of the message and the bandwidth of the channel.

$$ \begin{array}{rcl} TT & = & \displaystyle \frac{MS}{B} \\\ \sf transmission\ time & = & \sf \frac{message\ size}{bandwidth} \end{array} $$

Queueing time

The time needed for each intermediate or end device to hold the message before it can be processed.