# Ece 5625 communication systems i 3 107 chapter 3

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ECE 5625 Communication Systems I 3-107
CHAPTER 3. ANALOG MODULATION where d n = ± 1 represents random data bits and p ( t ) is a pulse shaping function, say p ( t ) = 1 , 0 t T 0 , otherwise Note that in this case m 2 ( t ) = 1, so there is a strong DC value present 2 4 6 8 10 1 0.5 0.5 1 2 4 6 8 10 1 0.5 0.5 1 t / T t / T m ( t ) m ( t )cos( ω c t ) BPSK modulation Digital signal processing techniques are particularly useful for building PLLs In the discrete-time domain, digital communication waveforms are usually processed at complex baseband following some form of I-Q demodulation 3-108 ECE 5625 Communication Systems I
3.4. FEEDBACK DEMODULATORS LPF A/D A/D LPF 0 o -90 o x IF ( t ) cos[2 π f cL t + φ L ] f s f s Sampling clock Discrete- Time r [ n ] = r I [ n ] + jr Q [ n ] r I ( t ) r Q ( t ) IF to discrete-time complex baseband conversion To Symbol Synch [ ] y n [ ] x n From Matched Filter NCO LUT - 1 z - 1 z [ ] v n Error Generation - 2 1 () 2 M M Im() p k a k Loop Filter [ ] e n e - j ˆ θ [ n ] ˆ θ [ n ] M th-power digital PLL (DPLL) carrier phase tracking loop ρ [ n ] φ [ n ] From Matched Filter To Symbol Synch ( ) j e ( ) j e M Rect. to Polar [ ] x n [ ] y n 1 arg() M L -Tap MA FIR L -Tap Delay () F ˆ θ [ n ] Non-Data Aided (NDA) feedforward carrier phase tracking ECE 5625 Communication Systems I 3-109
CHAPTER 3. ANALOG MODULATION 3.5 Sampling Theory We now return to text Chapter 2, Section 8, for an introduc- tion/review of sampling theory Consider the representation of continuous-time signal x ( t ) by the sampled waveform x δ ( t ) = x ( t ) n = −∞ δ ( t nT s ) = n = −∞ x ( nT s ) δ ( t nT s ) x ( t ) x δ ( t ) t t 0 0 T s -T s 2 T s 3 T s 4 T s 5 T s Sampling How is T s selected so that x ( t ) can be recovered from x δ ( t ) ? Uniform Sampling Theorem for Lowpass Signals Given F { x ( t ) } = X ( f ) = 0 , for f > W then choose T s < 1 2 W or f s > 2 W ( f s = 1 / T s ) to reconstruct x ( t ) from x δ ( t ) and pass x δ ( t ) through an ideal LPF with cutoff frequency W < B < f s W 2 W = Nyquist frequency f s / 2 = folding frequency 3-110 ECE 5625 Communication Systems I
3.5. SAMPLING THEORY proof : X δ ( f ) = X ( f ) f s n = −∞ δ ( f nf s ) but X ( f ) δ ( f nf s ) = X ( f nf s ) , so X δ ( f ) = f s n = −∞ X ( f nf s ) -W W -W W 0 0 f f X ( f ) X 0 X δ ( f ) - f s f s X 0 f s f s - W Guard band = f s - 2 W Lowpass reconstruction filter . . . . . . f -2 f s f s -f s 2f s X 0 f s . . . . . . Aliasing f s < 2 W f s > 2 W Spectra before and after sampling at rate f s As long as f s W > W or f s > 2 W there is no aliasing (spectral overlap) ECE 5625 Communication Systems I 3-111
CHAPTER 3. ANALOG MODULATION To recover x ( t ) from x δ ( t ) all we need to do is lowpass filter the sampled signal with an ideal lowpass filter having cutoff frequency W < f cutoff < f s W In simple terms we set the lowpass bandwidth to the folding frequency, f s / 2 Suppose the reconstruction filter is of the form H ( f ) = H 0 f 2 B e j 2 π f t 0 we then choose W < B < f s W For input X δ ( f ) , the output spectrum is Y ( f ) = f s H 0 X ( f ) e j 2 π f t 0 and in the time domain y ( t ) = f s H 0 x ( t t 0 ) If the reconstruction filter is not ideal we then have to design the filter in such a way that minimal desired signal energy is re-
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