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 3-1 Pulse Amplitude Modulation Baseband and Bandpass Signals One-, Two-, and Multidimensional Signals QAM and Complex Equivalent Baseband Signals 3-2 Pulse Shaping and ISI-free 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 M-ary 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) M-ary 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 )] DSB-SC-AM (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 3-dB bandwidth equivalent bandwidth (BW of block spectrum with equal energy and const. amplitude as at f c ) first spectral zero (BW of main lobe) Time-bandwidth 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 one-dimensional 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 N-dim. signal space; each representing a signal waveform orthogonal signals M-ary 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 Two-dimensional Signals Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 17/39 Two-dimensional Signals Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 18/39

10 Two-dimensional Signals bi-orthogonal signals [7.15] binary antipodal PAM of the basis functions M =4-ary signals with equal energies add signal vectors with inverted polarities: Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 19/39 Two-dimensional Signals M =8-ary signals with equal energies M =8-ary signals with (two) different energies Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 2/39

11 Two-dimensional Bandpass Signals QAM important example for 2D-signals: (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 complex-valued equivalent baseband representation; no BW loss! (see Chapter 2) Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 21/39 Two-dimensional QAM Signals Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 22/39

12 Two-dimensional 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 Two-dimensional 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 (M-ary) for n-th waveform; m 1, 2,...,M {s μ (t)} MN μ=1... MN-ary set of waveforms Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 25/39 Multidimensional Signals OFDM: parallel transmission to enlarge symbol duration against inter-symbol-interference (ISI) g n (t) = 1 T e j2πnt/t w(t)... n-th 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, 16/64-QAM) the symbols s μ (t) are MN-ary (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 TB-product) 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 N-fold 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 signal-space concept a preview to Section 5-1 complete treatment requires theory of random processes Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 28/39

15 Correlation-type 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 Pulse-Shaping Filtering and pulse-shaping at transmitter: pulse-shaping to reduce signal bandwith at receiver filter out noise and interferences hence filtering is applied at both sides Example: Low-pass (RX) filter: introduces inter-symbol interference (ISI) Objective is ISI-free transmission Achieved by Nyquist filtering (e.g. root-raised-cos filter) Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 32/39

17 Nyquist Pulse-Shaping (cont d) TX signal with rectangular pulse RX signal after lowpass filter; eye-diagram 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 Eye-diagram (right) shows RX signal quality Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 33/39 Nyquist Pulse-Shaping (cont d) eye-diagram without noise; LP-filtered rect. pulse rectangular pulse used at TX impulse response of lowpass filter used at RX eye-opening equivalent system impulse response 5 inter-symbol interference (ISI) ISI reduces eye-opening time time Construction of the eye-diagram Lowpass filter introduces inter-symbol-interference (ISI) Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 34/39

18 Nyquist Pulse-Shaping (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 ISI-free transmission { C for k = (constant) h e (kt + τ) = for k Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 35/39 Nyquist Pulse-Shaping (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 ISI-free transmission but cannot be realized (infinite extent; not causal) Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 36/39

19 Nyquist Pulse-Shaping (cont d) r= r =.5 r=1 sinc FT of cos-pulse product r= r =.5 1 r= frequency; normalized to Nyquist bandwidth B N = 1/(2T) time; normalized to symbol period T Cosine roll-off: allow bandwidth extension by B N (1 + r) frequency: convolve rectangular with cos-pulse time: multiply sinc (upper) with Fourier transform of cos-pulse (center); product Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 37/39 Nyquist Pulse-Shaping (cont d) r= r =.5 r= r= r =.5 r= time; normalized to symbol period T time; normalized to symbol period T left-hand figure: equivalent system impulse responses h e (t) with cos-roll-off Nyquist filtering right-hand figure: received data sequences Fund. of Digital CommunicationsCh. 3: Digital Modulation p. 38/39

20 Nyquist Pulse-Shaping (cont d) r= r =.5 r= time; normalized to symbol period T time; normalized to symbol period T eye diagram (right-hand 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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