Comp 349/449 Week 2 Read: Chapter 2: 2.1, 2.3, 2.4, Appendix A Chapter 5: 5.1, 5.2 (skim), 5.3 ================================= On the train home, at one point I scanned 58 networks! (More precisely, that's how many showed up in the windows "Choose a wireless network" box; some of these may have accumulated during my walk to the train station.) Later direct counts: 9 (~ Oak Park) 11 21 39 at Oak Park train stop (except one was another train rider, ad hoc) Maybe they sit in the cache for a while?? Remember: there are only three useable channels! =========================================================================== =========================================================================== Decibels Appendix 2A: Decibels are a logarithmic RELATIVE unit. dB = 10 log_10 (P2/P1), where P2 and P1 are two power levels being compared. 3 dB = 2x log 2 = 0.30103 5 dB = 3x log 3 = 0.477 10db = 10x 20db = 100x Common uses: to measure GAIN: by how much a signal is increased to measure loss: coax loss of 8 dB / 100m to measure relative performance signal-to-noise ratio Sometimes used as an absolute, rather than relative, unit, by comparing with P1 = 1 watt. In this case we use the name dBW =========================================================================== =========================================================================== 2.1: Time-domain concepts, Frequency-domain concepts. Signal as a function of time Signal as a set of frequencies Notion of bandwidth: anything other than a pure sine wave decomposes into a SET of frequencies, additively. Typical case: modulating a 1GHz sine with, say, a ~1MHz signal produces a RANGE of frequencies: Ideally +/- 1 MHz: 0.999-1.001 GHz. But actual range can be quite a bit larger, too. Two ways to see this: 1. practical. Consider a 1.000 GHz signal switched on and off every microsecond (1 MHz rate) (1 on and 1 off every microsecond). Each "pulse" thus consists of 1000 cycles. It takes *time* to verify a frequency exactly. After 1 microsec, a 1.001GHz signal is a full cycle off from a 1.000 GHz signal. After half a microsec, the two signals are exactly 180 degrees out-of-phase. That can barely be detected as a mismatch; in general, with a switched signal like this there's no way to read the frequency exactly so as to rule out anything in the range 0.999-1.001 GHz 2. math. Consider a 1.000 GHz signal that is amplitude-modulated according to a 1 MHz signal. The formula for a sine wave with frequency f is sin(2*pi*f*t) So the AM signal, where the strength of the 1GHz signal varies in proportion to the 1MHz signal, is the product of these (sort of): sin(2*pi*10^9*t)*sin(2*pi*10^6*t) Using the sin/cos identity sin A sin B = (1/2)cos (A-B) - (1/2)cos(A+B) this is (1/2) cos (2*pi*0.999*10^9*t) - (1/2) cos(2*pi*1.001*10^9*t) That is, two cosine waves at frequencies 0.999GHz and 1.001GHz. (cosine waves are the same as sine waves, except shifted 90 degrees.) sin v cosine, and PHASE Bottom line: all real signals have band WIDTH: carrier frequencies must have some SEPARATION, or they cannot carry data. ======================================= Basic Fourier Analysis example: sum of sines converging to a square wave All periodic signals, with period L, can be expressed as a "trigonometric polynomial" of period L: p = pi A1 sin 2pLt + A2 sin 2*2pLt + A3 sin 3*2pLt + ... + B1 cos 2pLt + B2 cos 2*2pLt + B3 cos 3*2pLt + ... Fig 2.5: band-width of various signals =========================================================================== Section 2.3: Nyquist and Shannon limits Nyquist: Given a (frequency) bandwidth B, max data capacity for binary transmission is 2B bits/sec. If we have M possible levels we can do better. M levels determine log2(M) bits. So our data rate becomes C = 2B * log2(M) Practical limit on M is noise. ==== Shannon formula: limits C in terms of SNR : signal-to-noise ratio SNR = B log2(1 + SNR) See example 2.1: given a bandwidth and snr, use Shannon's formula to get best C, and then use Nyquist's formula to solve for M. =========================================================================== =========================================================================== propagation: ground wave: works best in AM band and below (1 MHz). sky wave "reflection" or ionosphere reflection: works up through ~30 MHz (upper end of "short-wave" band) Above: line-of-sight, though refractive effects may still extend this slightly. =========================================================================== =========================================================================== 5.1: Antennas converts between electrical EM energy and radiated EM energy. -> transmit <- receive It is not necessarily true (in fact is often very untrue) that larger = better. since antennas perform best at sizes that have some integral relationship to wavelength (or quarter wavelengths). half-wave dipole, quarter-wave antenna Resonance and the Bay of Fundy http://en.wikipedia.org/wiki/Bay_of_Fundy Radiation/reception patterns of these. Beam width: angle between full-power direction and 50%-power direction Car antennas: FM band is ~ 100MHz. wavelength: 3 meters (3x10^8 m/sec * 10^-8 sec) or 300 m/microsec divided by 100 cycles/microsec Parabolic antenna: Table 5.1: at 12GHz (L = (30 cm/ns) / (12 cycles/ns) = 2.5 cm/cycle) diameter 50 cm = 20 L 3.5 degrees 100 cm = 40L 1.75 degrees 200 cm = 80L 0.875 degrees Antenna Gain: ratio of power in best direction to what it would be with an isotropic antenna. Table 5.2 of Stallings: some antenna gain factors. Note that parabolic gain is ~ 7 * area in square wavelengths Effective Area: area of a circle of radius 2: 4pi effective area measured in square wavelengths: G/4pi effective area: G*L^2/4pi Table 5.2 has effective area numbers, too. Note effective area of "standard" antennas is order-of-magnitude L^2. Also note effective area of parabola is about 56% of "real" area. ======================================== Yagi antenna: L = wavelength reflector dipole director ... director --> (not *exactly*!!) .55L .50L .45L spacing: approx .5 L --------------- Cantenna: 2km is not unheard of. Wavelength of 2.4GHz: 5" Optimal can dimensions: len: 1 wavelength (5") diameter: about 0.7 lambda, ~4" Yagi Cantenna (original Pringles cantenna): uses a yagi director with 0.2 L washers (1") ~15dB gain ======== waveguide cantenna very different principle than yagi: no washers! different can dimensions, too A waveguide tube would carry the signal arbitrarily far. Chopping the tube off means the signal is carried "into the ether", from whence it radiates. =============== FCC 15.247: rule on antenna gain operator safety: although 2.4 GHz is NOT a resonant frequency for water molecules, internal heating does occur (this IS the microwave frequency band, too), and operators should avoid excessive doses. ======================================================================== Antenna calculations Pt = transmitted power Pr = received power all antennas are perfect isotropic d = distance away Pr/Pt = L^2 / (4pi*d)^2 (inverted from book form) or Pr/Pt = (L/d)^2 / (4pi)^2 = 1/(4pi*D)^2, where D =d/L is distance in wavelengths Consequence: Loss is inverse-square Loss is much faster (by a constant) for higher wavelengths Ratio applies when d is big enough; when 4pi*d = L (d = L/4*pi) then formula says no loss. This is too close for accuracy though.