Wireless PHY: Modulation and Demodulation Y. Richard Yang 09/11/2012

2 Outline r Admin and recap r Amplitude demodulation r Digital modulation

3 Admin r Assignment 1 posted

4 r Objective o Frequency assignment r Basic concepts o the information source (also called baseband) o carrier o modulated signal Recap: Modulation

5 Recap: Amplitude Modulation (AM) r Block diagram r Time domain r Frequency domain

6 Recap: Demod of AM r Design option 1: multiply modulated signal by e -jfct, and then LPF r Design option 2: quadrature sampling

7 Example: Scanner r Setting: a scanner scans 128KHz blocks of AM radio and saves each block to a file. r For the example file m During scan, fc = 710K m LPF = 128K (one each side)

8 Exercise: Scanner r Requirements m Scan the block in a saved file to find radio stations and tune to each station (each AM station has 10 KHz) m Audio device requires 48K sample rate for playback

9 Remaining Hole: How to Design LPF r Frequency domain view freq B-B freq B-B

10 Design Option 1 freq B-B freq B-B compute freq zeroing out outband freq compute lower-pass time signal This is essentially how image compression works. Problem(s) of Design Option 1?

11 Design Option 2: Impulse Response Filters r GNU software radio implements filtering using m Finite Impulse Response (FIR) filters m Infinite Impulse Response (IIR) Filters m FIR filters are more commonly used r FIR/IIR is essentially online, streaming algorithms r They are used in networks/communications/vision/robotics…

12 FIR Filter r An N-th order FIR filter h is defined by an array of N+1 numbers: r They are often stored backward (flipped) r Assume input data stream is x0, x1, …, h0h0 h1h1 h2h2 hNhN …

13 FIR Filter xnxn x n-1 x n-2 x n-3 h0h0 h1h1 h2h2 h3h3 **** x n+1 compute y[n]: 3 rd -Order Filter

14 FIR Filter xnxn x n-1 x n-2 x n-3 h0h0 h1h1 h2h2 h3h3 **** x n+1 compute y[n+1]

15 FIR Filter is also called convolution between x (as a vector) and h (as a vector), denoted as

16 Key Question Using h to Implement LPF r Q: m How to determine h? r Approach: m Understand the effects of y=g*h in the frequency domain

g*h in the Continuous Time Domain 17 r Remember that we consider x as samples of time domain function g(t) on [0, 1] and (repeat in other intervals) r We also consider h as samples of time domain function h(t) on [0, 1] (and repeat in other intervals) for (i = 0; i< N; i++) y[t] += h[i] * g[t-i];

Visualizing g*h 18 g(t) h(t) time 0 T 0 T

Visualizing g*h 19 g(t) h(0) time t 0 T 0 T g(t)

Fourier Series of y=g*h 20

Fubini’s Theorem r In English, you can integrate m first along y and then along x m first along x and then along y m at (x, y) grid They give the same result 21 See

Fourier Series of y=g*h 22

Summary of Progress So Far 23 y = g * h => Y[k] = G[k] H[k] In the case of Fourier Transform, y = g * h => Y[f] = G[f] H[f] is called the Convolution Theorem, an important theorem.

Applying Convolution Theorem to Design LPF 24 r Choose h() so that m H() is close to a rectangle shape m h() has a low order (why?) f 1/2-1/2 1

Sinc Function 25 r The h() is often related with the sinc(t)=sin(t)/t function f 1/2-1/2 1

FIR Design in Practice 26 r Compute h m MATLAB or other design software m GNU Software radio: optfir (optimal filter design) m GNU Software radio: firdes (using a method called windowing method) r Implement filter with given h m freq_xlating_fir_filter_ccf or m fir_filter_ccf

LPF Design Example 27 r Design a LPF to pass signal at 1 KHz and block at 2 KHz

LPF Design Example 28 #create the channel filter # coefficients chan_taps = optfir.low_pass( 1.0, #Filter gain 48000, #Sample Rate 1500, #one sided mod BW (passband edge) 1800, #one sided channel BW (stopband edge) 0.1, #Passband ripple 60) #Stopband Attenuation in dB print "Channel filter taps:", len(chan_taps) #creates the channel filter with the coef found chan = gr.freq_xlating_fir_filter_ccf( 1, # Decimation rate chan_taps, #coefficients 0, #Offset frequency - could be used to shift 48e3) #incoming sample rate

29 Outline r Recap r Amplitude demodulation m frequency shifting m low pass filter r Digital modulation

30 r Modulation of digital signals also known as Shift Keying r Amplitude Shift Keying (ASK): m vary carrier amp. according to data r Frequency Shift Keying (FSK) o vary carrier freq. according to bit value r Phase Shift Keying (PSK) o vary carrier freq. according to data 101 t 101 t 101 t Modulation

31 r BPSK (Binary Phase Shift Keying): m bit value 1: cosine wave cos(2πf c t) m bit value 0: inverted cosine wave cos(2πf c t+π) m very simple PSK r Properties m robust, used e.g. in satellite systems Q I 10 Phase Shift Keying: BPSK one bit time T 1 0

32 Phase Shift Keying: QPSK Q I r QPSK (Quadrature Phase Shift Keying): m 2 bits coded at a time m we call the two bits as one symbol m symbol determines shift of cosine wave m often also transmission of relative, not absolute phase shift: DQPSK - Differential QPSK

33 r Quadrature Amplitude Modulation (QAM): combines amplitude and phase modulation r It is possible to code n bits using one symbol m 2 n discrete levels Q I 0010 φ a Quadrature Amplitude Modulation r Example: 16-QAM (4 bits = 1 symbol) r Symbols 0011 and 0001 have the same phase φ, but different amplitude a and 1000 have same amplitude but different phase

Generic Representation of Digital Keying (Modulation) r Sender sends symbols one-by-one r M signaling functions g 1 (t), g 2 (t), …, g M (t), each has a duration of symbol time T r Each value of a symbol has a signaling function 34

Exercise: g i () for BPSK 35 r 1: m g 1 (t) = cos(2πf c t) t in [0, T] r 0: m g 0 (t) = -cos(2πf c t) t in [0, T] r Are the two signaling functions independent? m Hint: think of the samples forming a vector, if it helps, in linear algebra m Ans: No. g 1 (t) = -g 0 (t) cos(2πf c t)[0, T] 1 Q I 10 g 1 (t)g 0 (t)

Exercise: Signaling Functions g i () for QPSK 36 r 11: m cos(2πf c t + π/4) t in [0, T] r 10: m cos(2πf c t + 3π/4) t in [0, T] r 00: m cos(2πf c t - 3π/4) t in [0, T] r 01: m cos(2πf c t - π/4) t in [0, T] r Are the four signaling functions independent? m Ans: No. They are all linear combinations of sin(2πf c t) and cos(2πf c t). Q I

QPSK Signaling Functions as Sum of cos(2πf c t), sin(2πf c t) 37 r 11: cos(π/4 + 2πf c t) t in [0, T] -> cos(π/4) cos(2πf c t) + -sin(π/4) sin(2πf c t) r 10: cos(3π/4 + 2πf c t) t in [0, T] -> cos(3π/4) cos(2πf c t) + -sin(3π/4) sin(2πf c t) r 00: cos(- 3π/4 + 2πf c t) t in [0, T] -> cos(3π/4) cos(2πf c t) + sin(3π/4) sin(2πf c t) r 01: cos(- π/4 + 2πf c t) t in [0, T] -> cos(π/4) cos(2πf c t) + sin(π/4) sin(2πf c t) sin( 2πf c t ) cos( 2πf c t ) [cos(π/4), sin(π/4)] 01 [cos(3π/4), sin(3π/4)] [cos(3π/4), -sin(3π/4)] [-sin(π/4), cos(π/4)] We call sin(2πf c t) and cos(2πf c t) the bases.

38 Outline r Recap r Amplitude demodulation m frequency shifting m low pass filter r Digital modulation m modulation m demodulation

Key Question: How does the Receiver Detect Which g i () is Sent? 39 r Assume synchronized (i.e., the receiver knows the symbol boundary).

Starting Point 40 r Considered a simple setting: sender uses a single signaling function g(), and can have two actions m send g() or m nothing (send 0) r How does receiver use the received sequence x(t) in [0, T] to detect if sends g() or nothing?

Design Option 1 41 r Sample at a few time points (features) to check r Issue m Not use all data points, and less robust to noise

Design Option 2 42 r Streaming algorithm, using all data points in [0, T] m As each sample x i comes in, multiply it by a factor h T-i-1 and accumulate to a sum y m At time T, makes a decision based on the accumulated sum at time T: y[T] xTxT x2x2 x1x1 x0x0 h0h0 h1h1 h2h2 hThT ****

Example Streaming (Convolution/Correlation): r Assume incoming x is a rectangular pulse (in baseband) and h is also a rectangular pulse r A gif animation (play in ppt) presentation): m redline g(): the sliding filter h(t) m blue line f(): the input x() 43 Source:

Determining the Best h 44 where w is noise, Design objective: maximize peak pulse signal- to-noise ratio

Determining the Best h 45 Assume Gaussian noise, one can derive Using Fourier Transform and Convolution Theorem:

Determining the Best h 46 Apply Schwartz inequality By considering

Determining the Best h 47

Determining Best h to Use 48 xTxT x2x2 x1x1 x0x0 gTgT g2g2 g1g1 g0g0 **** xTxT x2x2 x1x1 x0x0 h0h0 h1h1 h2h2 hThT ****

Matched Filter Decision is called Matched filter. Example 49 decision time

Backup Slides 50

51 Modulation