Outline. Wireless PHY: Modulation and Demodulation. Recap: Modulation. Admin. Recap: Demod of AM. Page 1. Recap: Amplitude Modulation (AM)
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1 Outline Wireless PHY: Modulation and Demodulation Admin and recap Amplitude demodulation Digital modulation Y. Richard Yang 9// Admin Assignment posted Recap: Modulation Objective o Frequency assignment Basic concepts o the information source (also called baseband) o carrier o modulated signal baseband carrier Modulator Modulated signal 3 4 Recap: Amplitude Modulation (AM) Recap: Demod of AM Block diagram x(t) m x + A c cos c t Time me domain Domain x AM (t)=a c [+mx(t)]cos c t Design option : multiply modulated signal by e -jfct, and then LPF Design option : quadrature sampling Frequency Domain domain X(f) X AM (f) sideba -f m f m f -f c f c f 5 6 Page
2 Example: Scanner Setting: a scanner scans 8KHz blocks of AM radio and saves each block to a file. For the example file During scan, fc = 7K LPF = 8K (one each side) Exercise: Scanner Requirements Scan the block in a saved file to find radio stations and tune to each station (each AM station has KHz) Audio device requires 48K sample rate for playback 7 8 Remaining Hole: How to Design LPF Design Option Frequency domain view compute freq -B B freq compute lower-pass time signal -B B zeroing out outband freq freq -B B freq 9 This is essentially how image compression works. Problem(s) of Design Option? -B B freq Design Option : mpulse Response Filters FR Filter GNU software radio implements filtering using Finite mpulse Response (FR) filters nfinite mpulse Response (R) Filters FR filters are more commonly used FR/R is essentially online, streaming algorithms They are used in networks/ communications/vision/robotics An N-th order FR filter h is defined by an array of N+ numbers: They are often stored backward (flipped) h N h = [h, h,..., h N ] h h h Assume input data stream is x, x,, Page
3 FR Filter FR Filter x n-3 x n- x n- x n x n+ x n-3 x n- x n- x n x n+ 3 rd -Order Filter h 3 h h h h 3 h h h compute y[n]: y n = x n h + x n h x n N h N N = x n i h i 3 i= compute y[n+] 4 FR Filter Key uestion Using h to mplement LPF y n = x n h + x n h x n N h N is also called convolution between x (as a vector) and h (as a vector), denoted as : How to determine h? Approach: Understand the effects of y=g*h in the frequency domain y n = x n * h n 5 6 g*h in the Continuous Time Domain Visualizing g*h Remember that we consider x as samples of time domain function g(t) on [, ] and (repeat in other intervals) We also consider h as samples of time domain function h(t) on [, ] (and repeat in other intervals) for (i = ; i< N; i++) y[t] += h[i] * g[t-i]; y(t) = h(τ )g(t τ )dτ 7 g(t) h(t) T T time 8 Page 3
4 Visualizing g*h Fourier Series of y=g*h y(t) = h(τ )g(t τ )dτ g(t) g(t) t h() time Y[k] = y(t)e jπkt dt = h(τ )g(t τ )dτ $% '( e jπkt dt = h(τ )g(t τ )e jπkt dτ $% '( dt T T 9 Fubini s Theorem Fourier Series of y=g*h y(t) = h(τ )g(t τ )dτ n English, you can integrate first along y and then along x first along x and then along y at (x, y) grid They give the same result See Y[k] = h(τ )g(t τ )e jπkt dτ $% '( dt = h(τ )g(t τ )e jπkt dt $% '( dτ = h(τ ) g(t τ )e jπkt dt $% '( dτ = h(τ )e jπkτ g(t τ )e jπk(t τ ) dt $% '( dτ = h(τ )e jπkτ G[k]dτ = G[k]H[k] Summary of Progress So Far Applying Convolution Theorem to Design LPF y = g * h => Y[k] = G[k] H[k] Choose h() so that H() is close to a rectangle shape n the case of Fourier Transform, y = g * h => Y[f] = G[f] H[f] h() has a low order (why?) -/ / f is called the Convolution Theorem, an important theorem. 3 4 Page 4
5 Sinc Function The h() is often related with the sinc(t)=sin(t)/t function FR Design in Practice Compute h MATLAB or other design software GNU Software radio: optfir (optimal filter design) GNU Software radio: firdes (using a method called windowing method) sin(πt) e jπ ft = rect( f ) πt -/ / f 5 mplement filter with given h freq_xlating_fir_filter_ccf or fir_filter_ccf 6 LPF Design Example LPF Design Example Design a LPF to pass signal at KHz and block at KHz create the channel filter coefficients chan_taps = optfir.low_pass(., Filter gain 48, Sample Rate 5, one sided mod BW (passband edge) 8, one sided channel BW (stopband edge)., Passband ripple 6) 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(, Decimation rate chan_taps, coefficients, Offset frequency - could be used to shift 48e3) incoming sample rate 7 8 Outline Recap Amplitude demodulation frequency shifting low pass filter Digital modulation Modulation Modulation of digital signals also known as Shift Keying Amplitude Shift Keying (ASK): vary carrier amp. according to data Frequency Shift Keying (FSK) o vary carrier freq. according to bit value t t Phase Shift Keying (PSK) o vary carrier freq. according to data t 9 3 Page 5
6 Phase Shift Keying: BPSK Phase Shift Keying: PSK BPSK (Binary Phase Shift Keying): bit value : cosine wave cos(πf c t) bit value : inverted cosine wave cos(πf c t+π) very simple PSK Properties robust, used e.g. in satellite systems one bit time T one bit time T PSK (uadrature Phase Shift Keying): bits coded at a time we call the two bits as one symbol symbol determines shift of cosine wave often also transmission of relative, not absolute phase shift: DPSK - Differential PSK 3 3 uadrature Amplitude Modulation uadrature Amplitude Modulation (AM): combines amplitude and phase modulation t is possible to code n bits using one symbol n discrete levels a φ Example: 6-AM (4 bits = symbol) Symbols and have the same phase φ, but different amplitude a. and have same amplitude but different phase Generic Representation of Digital Keying (Modulation) Sender sends symbols one-by-one M signaling functions g (t), g (t),, g M (t), each has a duration of symbol time T Each value of a symbol has a signaling function Exercise: g i () for BPSK Exercise: Signaling Functions g i () for PSK : : g (t) = cos(πf c t) t in [, T] g (t) = -cos(πf c t) t in [, T] Are the two signaling functions independent? Hint: think of the samples forming a vector, if it helps, in linear algebra Ans: No. g (t) = -g (t) g (t) - g (t) cos(πf c t)[, T] 35 : : cos(πf c t + π/4) t in [, T] cos(πf c t + 3π/4) t in [, T] : cos(πf c t - 3π/4) t in [, T] : cos(πf c t - π/4) t in [, T] Are the four signaling functions independent? Ans: No. They are all linear combinations of sin(πf c t) and cos(πf c t). 36 Page 6
7 PSK Signaling Functions as Sum of cos(πf c t), sin(πf c t) Outline : cos(π/4 + πf c t) t in [, T] -> cos(π/4) cos(πf c t) + -sin(π/4) sin(πf c t) : cos(3π/4 + πf c t) t in [, T] -> cos(3π/4) cos(πf c t) + -sin(3π/4) sin(πf c t) : cos(- 3π/4 + πf c t) t in [, T] -> cos(3π/4) cos(πf c t) + sin(3π/4) sin(πf c t) : cos(- π/4 + πf c t) t in [, T] -> cos(π/4) cos(πf c t) + sin(π/4) sin(πf c t) [cos(3π/4), sin(3π/4)] [cos(3π/4), -sin(3π/4)] sin(πf c t) [cos(π/4), sin(π/4)] cos(πf c t) [-sin(π/4), cos(π/4)] Recap Amplitude demodulation frequency shifting low pass filter Digital modulation modulation demodulation We call sin(πf c t) and cos(πf c t) the bases Key uestion: How does the Receiver Detect Which g i () is Sent? Assume synchronized (i.e., the receiver knows the symbol boundary). Starting Point Considered a simple setting: sender uses a single signaling function g(), and can have two actions send g() or nothing (send ) How does receiver use the received sequence x(t) in [, T] to detect if sends g() or nothing? 39 4 Design Option Sample at a few time points (features) to check ssue Not use all data points, and less robust to noise Design Option Streaming algorithm, using all data points in [, T] As each sample x i comes in, multiply it by a factor h T-i- and accumulate to a sum y x x x x T h T h h h At time T, makes a decision based on the accumulated sum at time T: y[t] 4 4 Page 7
8 Example Streaming (Convolution/Correlation): Assume incoming x is a rectangular pulse (in baseband) and h is also a rectangular pulse A gif animation: redline g(): the sliding filter h(t) blue line f(): the input x() Determining the Best h y = (g + w)* h = g*h + w * h = g o + n where w is noise, g o (t) = g*h n = w * h Design objective: maximize peak pulse signalto-noise ratio Source: g o (t) = g*h Determining the Best h Assume Gaussian noise, one can derive E[n (T )] = N H( f ) df Using Fourier Transform and Convolution Theorem: g o (T ) = G ( f )e jπ ft df = η = G( f )H( f )e jπ ft df N H( f ) df G( f )H( f )e jπ ft df 45 Determining the Best h Apply Schwartz inequality x( f )y( f )df x( f ) df By considering x( f ) = H( f ) y( f ) = G( f )e jπtf y( f ) df η = G( f )H( f )e jπ ft df N H( f ) df equal iff x( f ) = ky *( f ) H opt ( f ) = k[g( f )e jπ ft ]* jπ ft = kg *( f )e 46 Determining the Best h jπ ft H opt ( f ) = kg *( f )e η = G( f )H( f )e jπ ft df N H( f ) df (t) = H opt ( f )e jπ ft = kg *( f )e jπ ft e f = (t) = kg(t t) f = = kg( f )e jπ ft e f = = kg( f )e f = = kg( f )e f = jπ f (T t) jπ f (T t) jπ ft jπ ft 47 Determining Best h to Use x x x x T x x x x T h T h h h (t) = kg(t t) g g g g T 48 Page 8
9 Matched Filter Decision (t) = kg(t t) is called Matched filter. Example (t) = kg(t t) Backup Slides decision time 49 5 Modulation 5 Page 9
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