# Signal Properties

The equation below represents a typical sinusoidal signal with frequency represented in Hertz:

$v(t) = V_p sin (2\pi f t + \phi)$

• $v(t)$ is the signal
• $V_p$ is the peak amplitude of the signal
• $f$ is the frequency of the signal (in Hz)
• $t$ is time
• $\phi$ is the phase shift of the signal

The following sub-sections go in to a little more detail of the terms listed above.

## Amplitude

Amplitude is a measure of the magnitude or strength of a signal. In electrical circuits, the amplitude of a signal is typically measured in volts, but there are different “types” of amplitudes that are often reported, although they are all talking about the same thing.

Peak amplitude is the highest magnitude that a signal reaches (usually measured from 0), and it is often denoted as $V_p$  to denote Voltage Peak.

Peak-to-peak amplitude is a measurement from the lowest point of the signal to the highest point of the signal, and is often denoted as $V_(p-p)$ to denote Voltage Peak to Peak. When reporting the $V_p$  of a signal, it will be half the value of the $V_(p-p)$ of the same signal.

The Root Mean Square or (RMS) amplitude is another means of reporting amplitude.   For complex waveforms, especially non-repeating signals like noise, the RMS amplitude is usually used because it is unambiguous and because it has physical significance . RMS is calculated by examining the signal for a specific length of time (e.g., the period for a repeating signal). To calculate, the signal is first squared over all values in the portion of the signal you are examining, then the mean or average of those values is determined, finally the square root of that average is taken. The result is the RMS value of the signal.

### Frequency and Wavelength

Frequency is a measure of how quickly a repetitive signal repeats itself. To calculate the frequency of an event, the number of occurrences of the event within a fixed time interval are counted, and then divided by the length of the time interval.  For signals, this event corresponds to the signal reaching the peak amplitude. The units used for frequency are called Hertz which are abbreviated Hz.  A signal with a frequency of 1 Hz has a period of one second from one peak to the next.  In other words, the event that is being counted (the signal reaching peak amplitude) occurs once per second – giving a 1 Hz signal.

In communications, the rate of change (frequency) of a given signal is proportional to the amount of information that can be contained in that signal, so a signal with a frequency of 1 Hz cannot carry much information in a given time frame. Wireless standards such as 802.11 operate at much higher frequencies (the various flavours of 802.11 operate at around 2.4GHz or at around 5.8GHz, 1 GHz = 1,000,000,000 Hz).

RF signals propagate through the air on electromagnetic waves which travel at the speed of light. In a vacuum, the speed of light is approximately 300,000 km/sec, and is slower in a non-vacuum medium. Light traveling through the earth’s atmosphere is only marginally slower than 300,000 km/sec, so we generally use that speed as a first order approximation for the speed of light in the earth’s atmosphere.

Wavelength is the physical distance between one peak of a wave and the next.  Frequency, velocity and wavelength of electromagnetic signals are all intimately related by the function:

$\lambda = \frac{c}{f}$

• $\lambda$ is the wavelength of the signal in meters
• $c$ is the speed of light in the medium the signal is travelling in
• $f$ is the frequency of the signal (in Hz)

#### Examples

1) Visible light with a wavelength of 475nm is blue in colour. What is the frequency of this blue light?

$\lambda = \frac{c}{f} \rightarrow f = \frac{c}{\lambda} = \frac{3 \times 10^8 \frac{m}{s}}{475nm} = 6.3 \times 10^{14} Hz$

2) A local radio station transmits at a center frequency of 104.7 MHz.  What is the wavelength of the signals being transmitted?

$\lambda = \frac{c}{f} = \frac{3 \times 10^8 \frac{m}{s}}{104.7MHz} = 2.86m$

### Phase

Phase describes where in a cycle a wave is. It also describes the relative difference between where in the cycle two different waves of the same frequency currently are. When two signals of the same frequency reach the peaks and the valleys at the same time, they are considered “in-phase” with each other. When the reach the peaks and the valleys at different times, they are out of phase by some amount. That amount can be quantified by measuring the time between when the two signals reach their peaks. One entire cycle is 360º, so this phase difference measurement can be  between 0º and 360º (although 0 and 360 mean the same thing). The following picture shoes a reference signal followed by a signal that is in phase, a signal that is 180 degrees out of phase, and a signal that is 90 degrees out of phase.

### Polarization

Polarization is the orientation of the electromagnetic (EM) field of a signal as it propagates through its transmission medium. As you might expect, an electromagnetic field consists of an electric field (E field) and a magnetic field (H field or B field). As an EM signal travels, its E field and H field are changing sinuosoidally and are perpendicular to each other and perpendicular to the direction of propagation. You can see this in the following diagram (λ is the wavelength and the signal is propagating in the Z direction):

For dipole type antennas, the E field is oriented parallel to the antenna element (the antenna element is the source of the signal). The H field is oriented perpendicular to the antenna. Horizontal polarization occurs when the E field (and antenna) are oriented parallel to the ground while vertical polarization occurs when the E field (and antenna) are oriented perpendicular to the ground

Polarization is an important thing to consider in communication systems because a signal received at a receiving antenna will be strongest if its orientation lines up with the polarization of the transmitted signal.  To illustrate this idea, let’s look at two cases:

1. The transmit antenna is vertically oriented but the receive antenna is horizontally oriented. In this case, the receive antenna does not “see” much of the vertically polarized electric field of the signal, so the received signal strength is low. This sketch shows a vertically oriented transmit antenna and horizontally oriented receive antenna

2. The transmit antenna and the receive antenna are both vertically oriented so that the receiver “sees” all of the vertically polarized electric field of the transmitted signal so the received signal is strong. This sketch shows vertically oriented transmit and receive antennae