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

2. 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

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

4. Define: Solid angle, 
for a spherical surface

5. The total power radiated by an antenna is given by

6. is the normalized radiation intensity

7. 3 – D Pattern of a Narrow – Beam Antenna

8. 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)

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

10. Side lobes are undesirable
Wasted energy
Possible interference

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

12. 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

13. 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

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

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

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

17. 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

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

19. 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

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

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

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

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

24. 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

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

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

27. To find the radiation resistance:
Find the far – field power by integrating the far – field power density over a sphere
Equate to

28. 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

29. Solution
The parameters of copper are

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

32. Half – Wave Dipole Antenna
In phasor form:

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

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

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

36. 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

39. is max when

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

41. 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 

42. 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

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

44. 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?

45. 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

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

47. Friis transmission formula
On the receiving side,
Friis transmission formula

48. When the antennas are not aligned (More general expression)

49. 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

50. 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.

51. 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)

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

53. 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

54. 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

55. Determine all possible output frequencies
Homework
Determine all possible output frequencies

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

57. 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)

58. 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

59. 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

60. 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

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

62. Eff = 100 %
Eff = 33 %
Eff = 100 %

63. Modulation Index
Measure of the modulating signal wrt the carrier signal