# Propagation and Modulation of RF Waves

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Propagation and Modulation of RF Waves
ECE 480 Wireless Systems Lecture 4 Propagation and Modulation of RF Waves

Antenna pattern: Describes the far – field directional properties of an antenna when measured at a fixed distance from the antenna 3 – d plot that displays the strength of the radiated field (or power density) as a function of direction (spherical coordinates) specified by the zenith angle  and the azimuth angle  From reciprocity, a receiving antenna has the same directional antenna pattern as the pattern that it exhibits when operated in the transmission mode

The differential power through an elemental area dA is
always in the radial direction in the far – field region

Define: Solid angle,  for a spherical surface

The total power radiated by an antenna is given by

3 – D Pattern of a Narrow – Beam Antenna

Antenna Pattern It is convenient to characterize the variation of F ( , ) in two dimensions Two principle planes of the spherical coordinate system Elevation Plane ( - plane) Corresponds to a single value of  ( = x –z plane) ( = y –z plane) Azimuth Plane ( - plane) Corresponds to  = 90 o (x – y plane)

Clearer to express F in db for highly directive patterns
 = 0 plane

Side lobes are undesirable
Wasted energy Possible interference

Beam Dimensions Define: Pattern solid angle  p  p = Equivalent width of the main lobe For an isotropic antenna with F ( , ) = 1 in all directions:

Defines an equivalent cone over which all the radiation of the actual antenna is concentrated with equal intensity signal equal to the maximum of the actual pattern

The half – power (3 dB) beamwidth, , is defined as the angular width of the main lobe between the two angles at which the magnitude of F ( , ) is equal to half its peak value

F () is max at  = 90 o ,  2 = 135 0 ,  1 = 45 o ,  = 135 o – 45 o = 90 o

Null Beamwidth,  null Beamwidth between the first nulls on either side of the peak

Antenna Directivity  p = Pattern solid angle For an isotropic antenna,  p = 4  D = 1

D can also be expressed as
S iso = power density radiated by an isotropic antenna D = ratio of the maximum power density radiated by the antenna to the power density radiated by an isotropic antenna

For an antenna with a single main lobe pointing in the z direction:

Determine: The direction of maximum radiation Pattern solid angle directivity half – power beamwidth in the y-z plane for an antenna that radiates into only the upper hemisphere and its normalized radiation intensity is given by

Solution The statement in the upper hemisphere can be written mathematically as

a. The function is maximum when  = 0 b. The pattern solid angle is given by Polar plot of

c. d. The half – power by setting Polar plot of

Example – Directivity of a Hertzian Dipole
For a Hertzian dipole:

Antenna Gain P t = Transmitter power sent to the antenna P rad = Power radiated into space P loss = Power loss due to heat in the antenna = P t – P rad Define: Radiation Efficiency,   = 1 for a lossless antenna

Accounts for the losses in the antenna
Define: Antenna Gain, G Accounts for the losses in the antenna

Radiation Resistance P loss = Power loss due to heat in the antenna = P t – P rad

Find the far – field power by integrating the far – field power density over a sphere Equate to

Example – Radiation Resistance and Efficiency of a Hertzian Dipole
A 4 – cm long center – fed dipole is used as an antenna at 75 MHz. The antenna wire is made of copper and has a radius a = 0.4 mm. The loss resistance of a circular wire is given by Calculate the radiation resistance and the radiation efficiency of the dipole antenna

Solution The parameters of copper are

At 75 MHz:  This is a short dipole From before,

Half – Wave Dipole Antenna
In phasor form:

For a short dipole Expand these expressions to obtain similar expressions for the half – wave dipole

Consider an infinitesimal dipole segment of length dz excited by a current and located a distance from the observation point

The far field due to radiation by the entire antenna is given by
Two assumptions: (length factor)

Note that "s" appears in the equation twice – once for the distance away and once for the phase factor is not valid for the length factor If Q is located at the top of the dipole, the phase factor is which is not acceptable

is max when

Directivity of Half – Wave Dipole
Need P rad and S (R , )

Radiation Resistance of Half – Wave Dipole
Recall: for the short dipole (l = 4 cm) at 75 MHz R rad =  R loss =  For the half – wave dipole (l = 4 m) at 75 MHz R loss = 1.8 

Effective Area of a Receiving Antenna
Assume an incident wave with a power density of S i The effective area of the antenna, A e , is P int = Power intercepted by the antenna It can be shown: = Magnitude of the open – circuit voltage developed across the antenna

The power density carried by the wave is
For the short dipole

In terms of D: Valid for any antenna Example: Antenna Area The effective area of an antenna is 9 m 2. What is its directivity in db at 3 GHz?

Friis Transmission Formula
Assumptions: Each antenna is in the far – field region of the other Peak of the radiation pattern of each antenna is aligned with the other Transmission is lossless

For an isotropic antenna:
(ideal) In the practical case, In terms of the effective area A t of the transmitting antenna

Friis transmission formula
On the receiving side, Friis transmission formula

When the antennas are not aligned (More general expression)

Homework 1. Determine the following: a. The direction of maximum radiation b. Directivity c. Beam solid angle d. Half – power beamwidth in the x – z plane for an antenna whose normalized radiation intensity is given by: Hint: Sketch the pattern first

2. An antenna with a pattern solid angle of 1
2. An antenna with a pattern solid angle of 1.5 (sr) radiates 30 W of power. At a range of 1 km, what is the maximum power density radiated by the antenna? 3. The radiation pattern of a circular parabolic – reflector antenna consists of a circular major lobe with a half – power beamwidth of 2 o and a few minor lobes. Ignoring the minor lobes, obtain an estimate for the antenna directivity in dB.

Analog Modulation High frequencies require smaller antennas Modulation impresses a lower frequency onto a higher frequency for easier transmission The signal is modulated at the transmission end and demodulated at the receiving end Several basic types Amplitude modulation (AM) Frequency modulation (FM) Pulse code modulation (PCM) Pulse width modulation (PWM)

Amplitude Modulation Carrier wave – High frequency signal that transports the intelligence Signal wave – Low frequency signal that contains the intelligence

AM transmitter DC shifts the modulating signal Multiplies it with the carrier wave using a frequency mixer Mixer must be nonlinear Output is a signal with the same frequency as the carrier with peaks and valleys that vary in proportion to the strength of the modulating signal Signal is amplified and sent to the antenna

The mixer is usually a "square law" device, such as a diode or B – E junction of a transistor
Suppose that we apply the following signals to a square law device The output will be

Determine all possible output frequencies
Homework Determine all possible output frequencies

Advantages Simplicity Cost Disadvantages Susceptible to atmospheric interference (static) Narrow bandwidth (550 – 1500 KHz)

AM Receiver Tunable filter Envelope detector (diode) Capacitor is used to eliminate the carrier and to undo the DC shift Will generally include some form of automatic gain control (AGC)

Forms of Amplitude Modulation
In the most basic form, an AM signal in the frequency domain consists of The carrier signal Information at f c + f m (upper sideband) Information at f c - f m (lower sideband) (US and LS are mirror images) This wastes transmission power Carrier contains no information Information is all contained in only one of the sidebands

Frequently, in communications systems, the carrier and/or one of the sidebands is suppressed or reduced If only the carrier is reduced or suppressed, the process is called "Double – Sideband Suppressed (Reduced) Carrier" (DSSC or DSRC) If the carrier and one of the sidebands is suppressed or reduced, the process is called "Single – Sideband Suppressed (Reduced) Carrier" (SSSC or SSRC) Often, the carrier and one of the sidebands is totally suppressed. This process is simply called "Single Sideband" The carrier must be regenerated at the receiver end

Example Consider a carrier with a frequency  c Suppose we want to modulate the carrier with a signal The signal is amplitude – modulated by adding m(t) to C The expression for this signal is Expanding this expression

Take Fourier Transform
Convert to frequency domain by taking the Fourier Transform Take Fourier Transform = Unit impulse function

Eff = 100 % Eff = 33 % Eff = 100 %

Modulation Index Measure of the modulating signal wrt the carrier signal