Physical Layer Fundamentals
Contents
Frequency, Period, Phase
- Frequency - $ f $
- Period - $ T = 1/f $
- Phase - $ \phi $
- Amplitude - $ A $
- Wave - $ A \sin(2 \pi ft + \phi) $
I-Q Graph
The I-Q graph is a 2D representation of the phase and amplitude.
In-phase component $ I $ + Quadrature component $ Q $
$ \sin(2 \pi f t + \frac{\pi}{4} ) $
= $ \sin(2 \pi f t) \cos( \frac{\pi}{4}) + \cos(2 \pi f t) \sin (\frac{\pi}{4}) $
= $ \frac{1}{\sqrt{2}} \sin(2 \pi f t) + \frac{1}{\sqrt{2}} \sin (2 \pi f t) $
Wavelength
The distance occupied by one cycle of a wave
$ \lambda = vT $
$ \lambda f = v $
$ c = 3 \times 10^8 m/s $
$ \lambda = c/f $
e.g 2.5 Ghz to Wavelength
$ \lambda = \frac{300 m/\mu s}{2.5 \times 10^9} = 12 cm $
e.g 5mm to Frequency
$ f = \frac{300 m/\mu s}{0.005} = 60 GHz$
Time and Frequency Domain
- Time Domain
- Frequency Domain
Wireless Spectrum Allocation
- Maximise utilisation
- Adapt to market needs
- Fair licensing
- Promote competition
- Ensure spectrum availability
- Licensing
Decibels
The unit to measure power, loss, signal-to-noise ratio (SNR)
When measuring power, we have a very large scale (i.e. pico watts to kilo watts)
$ dB = 10 \log_{10}(\frac{P_1}{P_2}) $
- Path Loss / Attenuation
- $ P_1 $ - Transmit
- $ P_2 $ - Receive
- SNR
- $ P_1 $ - Signal
- $ P_2 $ - Noise
- Signal Power
- $ P_1 $ - Signal
- $ P_2 $ - Reference
$ dBm $ - reference to 1 milliwatt
$ dBW $ - reference to 1 watt
$ dBm = dBW + 30 $
When converting dBW to dBm, as 1 W = 1000 mW...
When multiplying the ratio by 1000, we can use the logarithm law $log(a \times b) = log(a) + log(b) $.
This allows us to split the ratio $log(1000 \times \frac{a}{b}) = log(1000) + log(\frac{a}{b}) $
When evaluated, $ log_{10}(1000) = 30 $
Coding
- Symbol - the smallest element of a signal that can be detected
- Baud rate (modulation rate / symbol rate) - the rate at which a symbol can change = $ \frac{1}{symbol\ duration} $
- Data rate - Bits per second
- $ bps = Bd \times \log_2 (M) $ for an M-ary signal
- For a binary signal, $ Bps = Bd $
Modulation
The digital version of modulation is called keying
- ASK - Amplitude Shift Keying
- FSK - Frequency Shift Keying
- PSK - Phase Shift Keying
- DPSK - Differential Phase Shift Keying - Compare to previous wave
0
- No difference1
- Difference
- QPSK - Quadrature Phase Shift Keying - Look at new phase offset
11
- 45 degrees10
- 135 degrees00
- 225 degrees01
- 315 degrees
- Note: We can visualise this with a constellation diagram (I-Q graph)
- DPSK - Differential Phase Shift Keying - Compare to previous wave
- QAM - Quadrature Amplitude and Phase Modulation
- 4-QAM - 2 bits per symbol
- 16-QAM - 4 bits per symbol
- n bits per symbol -> 2^n QAM
Channel Capacity
- Capacity - Maximum data rate (bps) for a channel
- Nyquist Theorem - In noiseless channel, error-free transmission can occur when the $ baud\ rate \le 2 \times bandwidth$
- $ Capacity = 2 \times bandwidth \times log_2(M) $
- M - number of levels
Shannon's Theory
When noise is present, the maximum data rate for error free communication, is $ B\ log_2 (1 + S/N) $
i.e a SNR of 4.77 dB is required to achieve 10 Mbps through a 5 MHz channel.
$ 10 \times 10^6 = (5 \times 10^6) \times log_2(1 + S/N) $
$ 2 = log_2(1 + S/N) $
$ 1 + S/N = 2^2 = 4 $
$ S/N = 3 $
$ 10log(S/N = 3) = 4.77 dB $
Hamming Distance
The number of different bits, when two sequences are compared.
Increasing the hamming distance between codewords increases the correction rate.
Hamming distance of 3 can detect up to 2 bits of errors, but can only correct one bit errors.
Multiple Access Methods
- TDMA - Time Division Multiple Access
- Clients are allocated a portion of time to communicate
- Only one client at a time
- FDMA - Frequency Division Multiple Access
- Clients are allocated to different frequency groups
- Using FFT, etc... we can extract data from certain frequencies
- CDMA - Code Division Multiple Access
- (CDMA) FHSS - Frequency Hopping Spread Spectrum
- Frequently change to different frequencies
- Transmitter and Receiver share a pseudo-random seed (code) which produces the next frequency to hop to.
- Hard to intercept
- Requires more bandwidth
- Requires time and frequency synchronisation
- (CDMA) DSSS - Direct Sequence Spread Spectrum
- Data XOR code
Doppler Shift
Moving radios will affect the frequency.
- Moving towards - frequency increase
- Moving away - frequency decrease
- Frequency difference = velocity / wavelength = $ \frac{v}{\lambda} $ = $ \frac{vf}{c} $
Doppler Spread
Two rays will be received (due to reflections)
- Doppler Spread = $ \frac{2vf}{c} $ = 2 $ \times $ doppler shift
- Spread will either add or cancel out as the receiver moves
- Coherence time - time during which the channel response is constant
- $ 1 / doppler\ spread $ = $ \frac{c}{2cf} $ = $ \frac{\lambda}{2v} $
Symbol should be designed to remain for the duration of the coherence time.
High QAM needed for high data rate.
Duplexing
Bidirectional communication.
- FDD - Frequency Division Duplexing (Full Duplex)
- Allocate different frequency ranges for Tx and Rx
- TDD - Time Division Duplexing (Half Duplex)
- Use the same frequency for Tx and Rx
- Synchronise time allocation