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Tele4653 l1

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    Tele4653 l1 Tele4653 l1 Presentation Transcript

    • TELE4653 Digital Modulation & Coding Fundamentals Wei Zhang w.zhang@unsw.edu.au School of Electrical Engineering and Telecommunications The University of New South Wales
    • Outline Introduction to Communications Lowpass (LP) and Bandpass (BP) Signals Signal Space Concepts Expansion of BP Signals TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.1/2
    • Modulation The information signal is a low frequency (baseband) signal. Examples: speech, sound, AM/FM radio The spectrum of the channel is at high frequencies. Therefore, the information signal should be translated to a higher frequency signal that matches the spectral characteristics of the communication channel. This is the modulation process in which the baseband signal is turned into a bandpass modulated signal. TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.3/2
    • Properties of FT (1) Linearity Property: If g(t) ⇔ G(f ), then c1 g1 (t) + c2 g2 (t) ⇔ c1 G1 (f ) + c2 G2 (f ). Dilation Property: If g(t) ⇔ G(f ), then 1 f g(at) ⇔ G . |a| a TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.4/2
    • Properties of FT (2) Conjugation Rule: If g(t) ⇔ G(f ), then g ∗ (t) ⇔ G∗ (−f ). Duality Property: If g(t) ⇔ G(f ), then G(t) ⇔ g(−f ). TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.5/2
    • Properties of FT (3) Time Shifting Property: If g(t) ⇔ G(f ), then g(t − t0 ) ⇔ G(f ) exp(−j2πf t0 ). Frequency Shifting Property: If g(t) ⇔ G(f ), then exp(j2πfc t)g(t) ⇔ G(f − fc ). TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.6/2
    • Properties of FT (4) Modulation Theorem: Let g1 (t) ⇔ G1 (f ) and g2 (t) ⇔ G2 (f ). Then g1 (t)g2 (t) ⇔ G1 (f ) G2 (f ), ∞ where G1 (f ) G2 (f ) = −∞ G1 (λ)G2 (f − λ)dλ. Convolution Theorem: Let g1 (t) ⇔ G1 (f ) and g2 (t) ⇔ G2 (f ). Then g1 (t) g2 (t) ⇔ G1 (f )G2 (f ), ∞ where g1 (t) g2 (t) = −∞ g1 (τ )g2 (t − τ )dτ . TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.7/2
    • Properties of FT (5) Correlation Theorem: Let g1 (t) ⇔ G1 (f ) and g2 (t) ⇔ G2 (f ). Then ∞ g1 (τ )g2 (t − τ )dτ ⇔ G1 (f )G∗ (f ). ∗ 2 −∞ Rayleigh’s Energy Theorem: Let g1 (t) ⇔ G1 (f ) and g2 (t) ⇔ G2 (f ). Then ∞ ∞ |g(t)|2 dt = |G(f )|2 df. −∞ −∞ Note that in the above formula, it is “=”, not “⇔”. TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.8/2
    • Lowpass Signals A lowpass, or baseband, signal is a signal whose spectrum is located around the zero frequency. The bandwidth of a real LP signal is W . TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.9/2
    • Bandpass Signals A bandpass signal is a real signal whose spectrum is located around some frequency ±f0 which is far from zero. Due to the symmetry of the spectrum, X+ (f ) has all the information that is necessary to reconstruct X(f ). ∗ X(f ) = X+ (f ) + X− (f ) = X+ (f ) + X+ (f ) (1) TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.10/2
    • Bandpass SignalsDenote x+ (t) the analytic signal of BP signal x(t). Then, x+ (t) = F −1 [X+ (f )] = F −1 [X(f )u−1 (f )] (2) −1 1 1 = x(t) F [u−1 (f )] = x(t) δ(t) + j (3) 2 2πt 1 j = x(t) + x(t), ˆ (4) 2 2where in (2) the unit step signal u−1 (f ) is used, in (3) 1Convolution Property is used, and in (4) x(t) = πt x(t) is the ˆHilbert transform of x(t).For details of Fourier Transform, please refer to Tables on pp.18-19 in textbook. TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.11/2
    • Bandpass SignalsDefine xl (t) the lowpass equivalent of x(t) whose spectrum isgiven by 2X+ (f + f0 ), i.e., Xl (f ) = 2X+ (f + f0 ). Then, xl (t) = F −1 [Xl (f )] = F −1 [2X+ (f + f0 )] = 2x+ (t)e−j2πf0 t = [x(t) + j x(t)] e−j2πf0 t ˆ − − − −using(4) (5)Alternatively, we can write x(t) = [xl (t)ej2πf0 t ]. (6)It expresses any BP signals in terms of its LP equivalent. TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.12/2
    • Bandpass SignalsWe can continue to write xl (t) = [x(t) cos(2πf0 t) + x(t) sin(2πf0 t)] ˆ + j [ˆ(t) cos(2πf0 t) − x(t) sin(2πf0 (t))] . x (7)For simplicity, we write xl (t) = xi (t) + jxq (t), where xi (t) = x(t) cos(2πf0 t) + x(t) sin(2πf0 t)] ˆ (8) xq (t) = x(t) cos(2πf0 t) − x(t) sin(2πf0 (t)) ˆ (9)Solving above equations for x(t) and x(t) gives ˆ x(t) = xi (t) cos(2πf0 (t)) − xq (t) sin(2πf0 (t)) (10) x(t) = xq (t) cos(2πf0 (t)) + xi (t) sin(2πf0 (t)) ˆ (11) TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.13/2
    • Bandpass SignalsIf we define the envelope and phase of x(t), denoted by r x (t)and θx (t), respectively, by rx (t) = x2 (t) + x2 (t) i q (12) xq (t) θx (t) = arctan (13) xi (t)we have xl (t) = xi (t) + jxq (t) = rx (t)ejθx (t) .Using (6), we have x(t) = [rx (t)ejθx (t) ej2πf0 t ] = rx (t) cos(2πf0 (t) + θx (t)). (14) TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.14/2
    • Mod/Demod of BP SignalsFIGURE 2.1-5 (a) is a modulator given by Eq. (6).FIGURE 2.1-5(b) is a modulator given by Eq. (10).FIGURE 2.1-5(c) is a general representation for a modulator.FIGURE 2.1-6 (a) is a demodulator given by Eq. (5).FIGURE 2.1-6(b) is a demodulator given by Eq. (7).FIGURE 2.1-6(c) is a general representation for a demodulator. TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.15/2
    • Vector Space ConceptsFor n-dimensional vectors v1 and v2 , n ∗ H Inner product: v1 , v2 = i=1 v1i v2i = v 2 v1 Orthogonal: v1 , v2 = 0 n 2 Norm: v = i=1 |vi | Triangle inequality: v1 + v2 ≤ v1 + v2 with equality if v1 = av2 for some positive real scalar a Cauchy-Schwarz inequality: | v1 , v2 | ≤ v1 · v2 with equality if v1 = av2 for some complex scalar a TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.16/2
    • Signal Space ConceptsFor two complex-valued signals x1 (t) and x2 (t), ∞ Inner product: x1 (t), x2 (t) = −∞ x1 (t)x∗ (t)dt 2 Orthogonal: x1 (t), x2 (t) = 0 ∞ 2 dt 1/2 √ Norm: x(t) = −∞ |x(t)| = Ex Triangle inequality: x1 (t) + x2 (t) ≤ x1 (t) + x2 (t) Cauchy-Schwarz inequality: | x1 (t), x2 (t) | ≤ x1 (t) · x2 (t) = E x1 E x2 TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.17/2
    • Orthogonal Expansion of SignalsTo construct a set of orthonormal waveforms from signalssm (t), m = 1, 2, · · · , K, we use Gram-Schmidt procedure: s1 (t) 1. φ1 = √ E1 γk (t) 2. φk (t) = √ Ek for k = 2, · · · , K,where k−1 γk (t) = sk (t) − cki φi (t) (15) i=1 ∞ cki = sk (t), φi (t) = sk (t)φ∗ (t)dt i (16) −∞ ∞ 2 Ek = γk (t)dt (17) −∞ TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.18/2
    • Orthogonal Expansion of SignalsOnce we have constructed the set of orthonormal waveforms{φn (t)} (m = 1, 2, · · · , M ), we may write N sm (t) = smn φn (t), m = 1, 2, · · · , M (18) n=1 TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.19/2
    • BP and LP Orthonormal BasisSuppose that {φnl (t)} constitutes an orthonormal basis for theset of LP signals {sml (t)}. We have sm (t) = {sml (t)ej2πf0 t }, m = 1, 2, · · · , M (19) N = smln φnl (t) ej2πf0 t (20) n=1 N = smln φnl (t)ej2πf0 t (21) n=1 √Define φn (t) = 2 φnl (t)ej2πf0 t and φn (t) = √− 2 φnl (t)ej2πf0 t . TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.20/2
    • BP and LP Orthonormal BasisDefine √ φn (t) = 2 φnl (t)ej2πf0 t (22) √ ˜n (t) = − 2 φnl (t)ej2πf0 t . φ (23)Substituting (22)-(23) into (21), we may have N (r) (i) smln smln ˜ sm (t) = φn (t) + φn (t) (24) n=1 2 2 (r) (i)where we have assumed that smln = smln + jsmln .Eq. (24) shows how a BP signal can be expanded in terms of thebasis used for expansion of its LP signal. TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.21/2
    • Gaussian RV The density function of a Gaussian RV X is 1 (x − µX )2 fX (x) = exp − 2 . 2 2πσX 2σX 2 For a special case when µX = 0 and σX = 1, it is called normalized Gaussian RV. Q-function, defined as ∞ 1 Q(x) = √ exp(−s2 /2)ds. 2π x Q-function can be viewed as the tail probability of the normalized Gaussian RV. TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.22/2
    • Random Process The random process X(t) is viewed as RV in term of time. At a fixed tk , X(tk ) is a RV. Autocorrelation of the random process is RX (t, s) = E[X(t)X ∗ (s)]. Wide-sense stationary requires: 1) the mean of the random process is a constant independent of time, and 2) the autocorrelation E[X(t)X ∗ (t − τ )] = RX (τ ) of the random process only depends upon the time difference τ , for all t and τ . TELE4653 - Digital Modulation & Coding - Lecture 1. March 1, 2010. – p.23/2