Fund. of Digital Communications Ch. 3: Digital Modulation


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1 Fund. of Digital Communications Ch. 3: Digital Modulation Klaus Witrisal Signal Processing and Speech Communication Laboratory Graz University of Technology November 26, 215 Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 1/39 Outline 31 Pulse Amplitude Modulation Baseband and Bandpass Signals One, Two, and Multidimensional Signals QAM and Complex Equivalent Baseband Signals 32 Pulse Shaping and ISIfree Transmission Signal Spectrum Nyquist Pulse Shaping Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 2/39
2 References and Figures Figures refer to Chapter 7 of J. G. Proakis and M. Salehi, Communication System Engineering, 2nd Ed., Prentice Hall, 22 J. G. Proakis and M. Salehi, Grundlagen der Kommunikationstechnik, 2. Aufl., Pearson, 24 (in German) References to figures denoted as [7.x] Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 3/ Pulse Amplitude Modulation (PAM) In practice: TX signal is a stream of symbols s(t) = i= s [i] (t it ) where s [i] (t) {s m (t)} M m=1 represents symbol s[i], taken from an Mary alphabet {s m (t)} Basic assumption: Consecutive symbols do not interfere Thus we can concentrate on one single symbol Symbol index i is dropped without loss of generality ; the TX signal is s(t) {s m (t)} M m=1 Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 4/39
3 Pulse Amplitude Modulation (PAM) Transmission of information through modulation of signal amplitude Signal shape g T (t) is tailored to channel Baseband signals for baseband channels [7.4] Binary antipodal modulation; selects amplitude of a pulse waveform g T (t) 1 ˆ= A for s 1 (t) =Ag T (t) ˆ= A for s 2 (t) = Ag T (t) Bit rate R b, bit interval T b (= symbol interval) R b = 1 T b Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 5/39 PAM, Baseband (cont d) Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 6/39
4 PAM, Baseband (cont d) Mary PAM [7.5] Usually: M =2 k for integer k; k... nb. bits/symbol Symbol interval: T = k/r b = kt b [7.6] (Set of) M signal waveforms [7.7] s m (t) =A m g T (t), for m {1, 2,..., M}, t T Pulse shape g T (t) determines signal spectrum [7.9] Energy (can vary for m {1, 2,..., M}) E m = T s 2 m(t)dt = A 2 m E g... energy of pulse g T (t) T g 2 T (t)dt = A2 me g Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 7/39 PAM, Baseband (cont d) Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 8/39
5 PAM, Baseband (cont d) Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 9/39 PAM, Bandpass (Passband) Bandpass signals for bandpass channels Carrier modulation [7.8] Multiplication of s m (t) by carrier cos(2πf c t) u m (t) =s m (t)cos(2πf c t)=a m g T (t)cos(2πf c t), for m {1, 2,..., M}, f c... carrier frequency (center frequency) in frequency domain [7.9]: U m (f) = A m 2 [G T (f f c )+G T (f + f c )] DSBSCAM (Dual sideband, suppressed carr. AM) Channel bandwidth 2W (doubled w.r.t. baseband!) Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 1/39
6 PAM, Baseband (cont d) Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 11/39 PAM, Baseband (cont d) Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 12/39
7 Definitions of Bandwidth Definitions Absolute bandwidth 3dB bandwidth equivalent bandwidth (BW of block spectrum with equal energy and const. amplitude as at f c ) first spectral zero (BW of main lobe) Timebandwidth product is constant! e.g. first zero of rectangular pulse: Interval T vs. first zero B z of its Fourier transform: B z =1/T, hence TB z =1 Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 13/39 Geometric Representation in Signal Space PAM Signals are onedimensional s m (t) =s m ψ(t) baseband: ψ(t) = 1 Eg g T (t), t T s m = E g A m, m {1, 2,..., M} bandpass: ψ(t) = 2 g T (t)cos2πf c t E g s m = E g /2A m, m {1, 2,..., M} Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 14/39
8 Geometric Representation (cont d) Euclidean distance d mn = s m s n 2 Energy of PAM signals (baseband) E m = s 2 m = E g A 2 m, m {1, 2,...M} e.g.: symmetric PAM [7.11] Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 15/39 Two (and Multidimensional) Signals Simultaneous PAM of two (or more) basis functions yields additional points in Ndim. signal space; each representing a signal waveform orthogonal signals Mary symbols are represented by N = M orthogonal waveforms (see [7.12] [7.14] for M =2) Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 16/39
9 Twodimensional Signals Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 17/39 Twodimensional Signals Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 18/39
10 Twodimensional Signals biorthogonal signals [7.15] binary antipodal PAM of the basis functions M =4ary signals with equal energies add signal vectors with inverted polarities: Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 19/39 Twodimensional Signals M =8ary signals with equal energies M =8ary signals with (two) different energies Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 2/39
11 Twodimensional Bandpass Signals QAM important example for 2Dsignals: (digital) QAM PAM of the orthogonal carriers u m (t) =A mc g T (t)cos2πf c t A ms g T (t) sin 2πf c t in geometric representation: u m (t) =s m1 ψ 1 (t)+s m2 ψ 2 (t), with ψ 1 (t) = 2/E g g T (t)cos2πf c t ψ 2 (t) = 2/E g g T (t) sin 2πf c t s m =[s m1,s m2 ] T =[ E s A mc, E s A ms ] T QAM signals have a complexvalued equivalent baseband representation; no BW loss! (see Chapter 2) Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 21/39 Twodimensional QAM Signals Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 22/39
12 Twodimensional Bandpass Signals QAM (cont d) Functional block diagram of a (digital) QAM modulator Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 23/39 Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 24/39
13 Multidimensional Signals Twodimensional case: M =2 k signals have been constructed in 2D Multidimensional case: (OPAM) construct N orthogonal signals define signal points in these dimensions N s μ (t) = a m,n g n (t) n=1 {g n (t)} N n=1... N orthogonal waveforms (basis) {a m,n R(or C)}... PAM (or QAM) symbols (Mary) for nth waveform; m 1, 2,...,M {s μ (t)} MN μ=1... MNary set of waveforms Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 25/39 Multidimensional Signals OFDM: parallel transmission to enlarge symbol duration against intersymbolinterference (ISI) g n (t) = 1 T e j2πnt/t w(t)... nth subcarrier at f = n/t s μ (t) = 1 N/2 1 T n= N/2 a m,n e j2πnt/t w(t) {g n (t)}... orthogonal subcarriers (Fourier basis) w(t)... window function (e.g. rectangular) {a m,n C}... QAM symbols (e.g. QPSK, 16/64QAM) the symbols s μ (t) are MNary (N subcarriers) Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 26/39
14 Multidimensional Signals Spread Spectrum/CDMA: few points in N 1 dimensional space (large TBproduct) divide T into N chip intervals T c = T/N modulate chip waveform with spreading code {c n } g T (t) = N 1 n= c n g c (t nt c ); s m (t) =a m g T (t) (PAM/QAM) chip wavef. g c (t) has Nfold bandwidth 1/T c = N/T g T (t) is a broadband pulse of duration T,BWN/T; i.e. it has N dimensions: N orthogonal sequences {c n } can be found for multiple access (CDMA); enhanced robustness Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 27/39 Optimum Demodulation (preview) Intuitive introduction to the demodulator using the signalspace concept a preview to Section 51 complete treatment requires theory of random processes Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 28/39
15 Correlationtype demodulator Channel: Additive white Gaussian noise (AWGN) is added r(t) =s m (t)+n(t) transmitted signal {s m (t)},m=1, 2,...M is represented by N basis functions {ψ k (t)},k =1, 2,...N received signal r(t) is projected onto these basis functions {ψ k (t)} T r(t)ψ k (t)dt = T r k = s mk + n k, [s m (t)+n(t)]ψ k (t) dt k =1, 2,..., N Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 29/39 Correlation demod. (cont d) [Proakis 22] Output vector in signal space: r = s m + n Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 3/39
16 Correlation demod. (cont d) Received signal N r(t) = s mk ψ k (t)+ = k=1 N n k ψ k (t)+n (t) k=1 N r k ψ k (t)+n (t) k=1 Correlator outputs r =[r 1,r 2,...r N ] T are sufficient statistik for the decision i.e.: there is no additional info in n (t) n (t) is part of n(t) that is not representable by {ψ k (t)} Interpretation of r: noise cloud in signal space Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 31/ Nyquist PulseShaping Filtering and pulseshaping at transmitter: pulseshaping to reduce signal bandwith at receiver filter out noise and interferences hence filtering is applied at both sides Example: Lowpass (RX) filter: introduces intersymbol interference (ISI) Objective is ISIfree transmission Achieved by Nyquist filtering (e.g. rootraisedcos filter) Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 32/39
17 Nyquist PulseShaping (cont d) TX signal with rectangular pulse RX signal after lowpass filter; eyediagram RX signal corrupted 1 by noise RX signal after 1lowpass filter time time Rectangular pulse at transmitter; noise added on channel; lowpass filter at receiver for noise reduction Eyediagram (right) shows RX signal quality Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 33/39 Nyquist PulseShaping (cont d) eyediagram without noise; LPfiltered rect. pulse rectangular pulse used at TX impulse response of lowpass filter used at RX eyeopening equivalent system impulse response 5 intersymbol interference (ISI) ISI reduces eyeopening time time Construction of the eyediagram Lowpass filter introduces intersymbolinterference (ISI) Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 34/39
18 Nyquist PulseShaping (cont d) PAM signal after receiver filter y(t) = i= a[i]h e (t it ) a[i] {a m } M m=1... PAM (or QAM) of symbol i i... symbol (= time) index h e (t) =g T (t) h c (t) h(t)... cascade of TX pulse, channel IR, and RX filter Condition for ISIfree transmission { C for k = (constant) h e (kt + τ) = for k Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 35/39 Nyquist PulseShaping (cont d) Transmission at minimum bandwidth B N (Nyquist BW) Assume: sampling frequency equals symbol rate f s =1/T sampled signal can represent signals up to B N = f s /2=1/(2T ) Consider a rectangular frequency response for H e (f) H e (f) = rect(f,b N ) F h e (t) = 1 T sinc(t/t ) fulfills condition for ISIfree transmission but cannot be realized (infinite extent; not causal) Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 36/39
19 Nyquist PulseShaping (cont d) r= r =.5 r=1 sinc FT of cospulse product r= r =.5 1 r= frequency; normalized to Nyquist bandwidth B N = 1/(2T) time; normalized to symbol period T Cosine rolloff: allow bandwidth extension by B N (1 + r) frequency: convolve rectangular with cospulse time: multiply sinc (upper) with Fourier transform of cospulse (center); product Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 37/39 Nyquist PulseShaping (cont d) r= r =.5 r= r= r =.5 r= time; normalized to symbol period T time; normalized to symbol period T lefthand figure: equivalent system impulse responses h e (t) with cosrolloff Nyquist filtering righthand figure: received data sequences Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 38/39
20 Nyquist PulseShaping (cont d) r= r =.5 r= time; normalized to symbol period T time; normalized to symbol period T eye diagram (righthand figure): (long) sequence of received, filtered data symbols superimposed in diagram over two symbol intervals Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 39/39
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