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Sampling PAM- Pulse Amplitude Modulation (Continued) Sampling PAM- Pulse Amplitude Modulation (continued) EELE445-14 Lecture 14 Sampling Properties will be looking at for: •Impulse Sampling •Natural Sampling •Rectangular Sampling 1 Figure 2–18 Impulse sampling. Impulse Sampling 2 Ts ∞ ∞ δ ()t = δ (t − nT ) ⇒ D + D cos(nω t +ϕ ) Ts ∑ s 0 ∑ n s n n=−∞ n=1 2π 1 2 ϕn = 0 ωs = D0 = Dn = Ts Ts Ts 1 δ ()t = []1+ 2cos(ω t) + 2cos(2ω t) + 2cos(3ω t) + Ts s s s L Ts 2 Impulse Sampling ws (t) ∞ δ ()t = δ (t − nT ) Ts ∑ s n=−∞ w(t) w (t) = w(t)δ ()t = []1+ 2cos(ω t) + 2cos(2ω t) + 2cos(3ω t) + s Ts s s s L Ts 1 ∞ Ws ( f ) = F[]ws (t) = ∑W ( f − nf s ) Ts n=−∞ Impulse Sampling- text ∞ w (t) = w(t) δ (t − nT ) s ∑n=−∞ s ∞ = w(nT )δ (t − nT ) eq2 −171 ∑n=−∞ s s subsituting , ∞ 1 w (t) = w(t) e jnωst s ∑n=−∞ Ts 1 ∞ W ( f ) = W ( f − nf ) s ∑n=−∞ s Ts 3 Impulse Sampling The spectrum of the impuse sampled signal is the spectrum of the unsampled signal that is repeated every fs Hz, where fs is the sampling frequency or rate (samples/sec). This is one of the basic principles of digital signal processing. Note: This technique of impulse sampling is often used to translate the spectrum of a signal to another frequency band that is centered on a harmonic of the sampling frequency, fs. If fs>=2B, (see fig 2-18), the replicated spectra around each harmonic of fs do not overlap, and the original spectrum can be regenerated with an ideal LPF with a cutoff of fs/2. Impulse Sampling Undersampling and aliasing. 4 Natural Sampling Generation of PAM with natural sampling (gating). Natural Sampling Duty cycle =1/3 5 Natural Sampling f Null at s = 3 f d s PAM and PCM • PAM- Pulse Amplitude Modulation: – The pulse may take any real voltage value that is proportional to the value of the original waveform. No information is lost, but the energy is redistributed in the frequency domain. • PCM- Pulse Code Modulation: – The original waveform amplitude is quantized with a resulting loss of information 6 Figure 3–4 Demodulation of a PAM signal (naturally sampled). nth Nyquist region recovery Comb Oscillator n=1,2,3…. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 Figure 3–5 PAM signal with flat-top sampling. Impulse sample and hold Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 7 Figure 3–6 Spectrum of a PAM waveform with flat-top sampling. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 Figure 3–7 PCM trasmission system. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 8 PCM Transmission • Negative: The transmission bandwidth of the PCM signal is much larger than the bandwidth of the original signal • Positive: The transmission range of a PCM signal may be extended with the use of a regenerative repeater. Figure 3–8 Illustration of waveforms in a PCM system. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 9 Figure 3–8 Illustration of waveforms in a PCM system. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 Illustration of waveforms in a PCM system. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 10 Illustration of waveforms in a PCM system. •The error signal, (quantization noise), is inversely proportional to the number of quantization levels used to approximate the waveform x(t) •Companding is used to improve the SQNR (signal to quantization noise ratio) for small amplitude x(T) signals Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 Figure 3–9 Compression characteristics (first quadrant shown). Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 11 Figure 3–9 Compression characteristics (first quadrant shown). Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 Figure 3–10 Output SNR of 8-bit PCM systems with and without companding. ⎡ S ⎤ ⎛Vpeak range ⎞ ⎢ ⎥ = 4.77 + 6.02n − 20log⎜ ⎟ uniform quantion ⎣ N ⎦dB ⎝ xrms ⎠ ⎡ S ⎤ ⎢ ⎥ = 4.77 + 6.02n − 20log[]ln()1+ μ μ − law companding ⎣ N ⎦dB ⎡ S ⎤ ⎢ ⎥ = 4.77 + 6.02n − 20log[]1+ ln()A A − law companding ⎣ N ⎦dB Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 12 Quantization Noise, Analog to Digital Converter-A/D EELE445 Lecture 16 26 Figure 3–7 PCM trasmission system. q(x) Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 13 Quantization xˆn = Q(x) V x(t) Δ 2 V V Δ = = M 2n−1 M = 2n Q(x) -V Quantization – Results in a Loss of Information x(t) 2 V V Δ Δ = = n−1 2 M 2 Q(x) Lost Information Δ − After sampling, x(t)=x 2 i xi ∈R After Quantization: Q(x) = xˆ, x ∈R 14 Quantization Noise Quantization function: Define the mean square distortion: q(x) = (x − Q(x))2 = ~x 2 and Δ x − Q(x) ≤ 2 Quantization Δ 2 V V 2 Δ = = 1 n−1 q2 = ∫ q2dq M 2 Δ −Δ 2 n Δ2 M = 2 = 12 ()V 2 = = P the quantization noise 3M 2 nq where M=2n, V is ½ the A/D input range, and n is the number of bits 15 Quantization Noise from the expectation operator: ~ Since X is a random variable, so are Xˆ and X So we can define the mean squared error (distortion) as : D = E[]d ()X , Xˆ = E[](X − Q(X ))2 ~ The pdf of the error is uniformly distributed : X = X − Q(X ) f (~x) 1 ⎧Δ Δ ~ Δ ~ ⎪ − ≤ x ≤ Δ f (x) = ⎨ 2 2 2 ~x ⎩⎪0 otherwise Δ − Δ 2 2 SQNR – Signal to Quantization Noise Ratio 16 SQNR – Signal to Quantization Noise Ratio Px may be found using: SQNR – Signal to Quantization Noise Ratio The distortion, or “noise”, is therefore: Where Px is the power of the input signal 17 SQNR – Linear Quantization 2 2 E[X ]≤ xmax The SQNR decreases as Px The input dynamic range 2 < 1 increases xmax U-Law Nonuniform PCM used to increase SQNR for given Px , xmax, and n U=255 U.S 18 A-Law Nonuniform PCM a=87.56 U.S u-Law v.s. Linear Quantization 8 bit Px is signal power Relative to full scale Px 19 Pulse Code Modulation, PCM, Advantage compared with analog systems • b is the number of bits • γ is (S/N)baseband Relative to full scale • PPM is pulse position modulation 20.
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