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**Propagation and Modulation of RF Waves**

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

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**Antenna Radiation Characteristics**

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

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**The differential power through an elemental area dA is**

always in the radial direction in the far – field region

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Define: Solid angle, for a spherical surface

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**The total power radiated by an antenna is given by**

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**is the normalized radiation intensity**

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**3 – D Pattern of a Narrow – Beam Antenna**

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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)

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**Clearer to express F in db for highly directive patterns**

= 0 plane

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**Side lobes are undesirable**

Wasted energy Possible interference

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

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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

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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

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

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

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

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**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

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

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**Example – Antenna Radiation Properties**

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

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

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

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

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**Example – Directivity of a Hertzian Dipole**

For a Hertzian dipole:

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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

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**Accounts for the losses in the antenna**

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

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

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**To find the radiation resistance:**

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

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**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

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Solution The parameters of copper are

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

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**Half – Wave Dipole Antenna**

In phasor form:

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

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

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**The far field due to radiation by the entire antenna is given by**

Two assumptions: (length factor)

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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

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is max when

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**Directivity of Half – Wave Dipole**

Need P rad and S (R , )

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**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

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**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

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**The power density carried by the wave is**

For the short dipole

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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?

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**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

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**For an isotropic antenna:**

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

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**Friis transmission formula**

On the receiving side, Friis transmission formula

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**When the antennas are not aligned (More general expression)**

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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

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**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.

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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)

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

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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

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**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

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**Determine all possible output frequencies**

Homework Determine all possible output frequencies

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

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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)

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**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

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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

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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

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**Take Fourier Transform**

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

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

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Modulation Index Measure of the modulating signal wrt the carrier signal

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Goal Derive the radar equation for an isolated target

Goal Derive the radar equation for an isolated target

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