TELE4653 Digital Modulation & Coding                     Fundamentals                         Wei Zhang                   ...
Outline Introduction to Communications Lowpass (LP) and Bandpass (BP) Signals Signal Space Concepts Expansion of BP Signal...
Modulation The information signal is a low frequency (baseband) signal. Examples: speech, sound, AM/FM radio The spectrum ...
Properties of FT (1) Linearity Property: If g(t) ⇔ G(f ), then            c1 g1 (t) + c2 g2 (t) ⇔ c1 G1 (f ) + c2 G2 (f )....
Properties of FT (2) Conjugation Rule: If g(t) ⇔ G(f ), then                         g ∗ (t) ⇔ G∗ (−f ). Duality Property:...
Properties of FT (3) Time Shifting Property: If g(t) ⇔ G(f ), then              g(t − t0 ) ⇔ G(f ) exp(−j2πf t0 ). Frequen...
Properties of FT (4) Modulation Theorem: Let g1 (t) ⇔ G1 (f ) and g2 (t) ⇔ G2 (f ). Then                  g1 (t)g2 (t) ⇔ G...
Properties of FT (5) Correlation Theorem: Let g1 (t) ⇔ G1 (f ) and g2 (t) ⇔ G2 (f ). Then                ∞                ...
Lowpass Signals A lowpass, or baseband, signal is a signal whose spectrum is located around the zero frequency. The bandwi...
Bandpass Signals A bandpass signal is a real signal whose spectrum is located around some frequency ±f0 which is far from ...
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 ...
Bandpass SignalsDefine xl (t) the lowpass equivalent of x(t) whose spectrum isgiven by 2X+ (f + f0 ), i.e., Xl (f ) = 2X+ (...
Bandpass SignalsWe can continue to write         xl (t) = [x(t) cos(2πf0 t) + x(t) sin(2πf0 t)]                           ...
Bandpass SignalsIf we define the envelope and phase of x(t), denoted by r x (t)and θx (t), respectively, by                ...
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).F...
Vector Space ConceptsFor n-dimensional vectors v1 and v2 ,                                    n        ∗        H    Inner...
Signal Space ConceptsFor two complex-valued signals x1 (t) and x2 (t),                                      ∞    Inner pro...
Orthogonal Expansion of SignalsTo construct a set of orthonormal waveforms from signalssm (t), m = 1, 2, · · · , K, we use...
Orthogonal Expansion of SignalsOnce we have constructed the set of orthonormal waveforms{φn (t)} (m = 1, 2, · · · , M ), w...
BP and LP Orthonormal BasisSuppose that {φnl (t)} constitutes an orthonormal basis for theset of LP signals {sml (t)}. We ...
BP and LP Orthonormal BasisDefine                             √                φn (t) =  2 φnl (t)ej2πf0 t                 ...
Gaussian RV The density function of a Gaussian RV X is                         1            (x − µX )2            fX (x) =...
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 ...
Tele4653 l1
Tele4653 l1
Tele4653 l1
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Tele4653 l1

  1. 1. TELE4653 Digital Modulation & Coding Fundamentals Wei Zhang w.zhang@unsw.edu.au School of Electrical Engineering and Telecommunications The University of New South Wales
  2. 2. 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
  3. 3. 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
  4. 4. 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
  5. 5. 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
  6. 6. 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
  7. 7. 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
  8. 8. 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
  9. 9. 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
  10. 10. 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
  11. 11. 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
  12. 12. 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
  13. 13. 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
  14. 14. 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
  15. 15. 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
  16. 16. 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
  17. 17. 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
  18. 18. 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
  19. 19. 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
  20. 20. 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
  21. 21. 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
  22. 22. 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
  23. 23. 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

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