ECE8700 Communications System Engineering Chapter 6 Lecture 4 Conversion of Analog Yimin D. Zhang, Ph.D. Waveforms into Coded Pulses Department of Electrical & Computer Engineering Villanova University Spring 2014 Y. D. Zhang ECE-8700 Spring 2014 2 Outlines Pulse Modulations Sampling Process … PAM . Sampling Theorem . Nyquist Rate Pulse-Amplitude Modulation (PAM) Sampling Continuous- Analog Pulse Quantization Wave Modulation Modulation . Uniform and non-uniform quantization Quantization . Quantization noise … PCM Coding Pulse-Code Modulation (PCM) Delta Modulation . Delta-Sigma Modulation Digital Pulse Modulation Line Codes Y. D. Zhang ECE-8700 Spring 2014 3 Y. D. Zhang ECE-8700 Spring 2014 4 Sampling Process: Sampling Process: Time-Domain Representations Frequency-Domain Representations G( f ) Ideally sampled signal of a continuous- 1 wave signal g(t) g(t) . Through periodic sampling W W f . Product of the continuous-wave signal t P( f ) fs ( f k fs ) and a sequence of impulses 2 f s k p(t) p(t) (t nTs ) n f 2 f S f S 0 f S 2 f S t g(t) g (t) Multiplication in the time domain yields convolution in 2T T 0 T 2T s s s s frequency domain. g() t gtpt () () gt () ( t nT ) g (t) s gt() gt () ( t nT ) n s Alias due to sampling n gnT()(s t nTs ) n t GfGfPffGfmffGffGfmf () ()*() s ( ss ) () s ( s ) 2Ts Ts 0 Ts 2Ts mm Ts : sampling period; fs =1/Ts : sampling rate m0 Y. D. Zhang ECE-8700 Spring 2014 5 Y. D. Zhang ECE-8700 Spring 2014 6 Sampling Process: Sampling Process: Frequency-Domain Representations Frequency-Domain Representations G( f ) 1 G( f ) 1 W W f W W f P( f ) f ( f k f ) 2 f s s s k P( f ) fs ( f k fs ) 2 f s k 2 f f 0 f 2 f f S S S S f 2 f S f S 0 f S 2 f S Iff s 2W , the replicas of G( f ) do not overlap. G ( f ) fs f 2 f S f S W 0 W f S 2 f S f S W Y. D. Zhang ECE-8700 Spring 2014 7 Y. D. Zhang ECE-8700 Spring 2014 8 Sampling Process: Sampling Theorem Sampling Process : Signal Recovery [Sampling Theorem] If G( f )=0 for f W (band limited signal) and fW s 2 (with sufficient sampling rate). Then, G( f ) can be obtained by properly filtering G( f ). Otherwise, aliasing phenomenon occurs. (a) Anti-alias filtered 2W: Nyquist rate; 1/(2W): Nyquist interval spectrum of an information-bearing signal. (b) Spectrum of instantaneously sampled version of the signal, assuming the use of a f 2W : Complete recovery of the original signal is possible. s sampling rate greater than the Nyquist rate. (c) Magnitude response of reconstruction filter. f s 2W (Undersampled): original signal cannot be completely recovered. Y. D. Zhang ECE-8700 Spring 2014 9 Y. D. Zhang ECE-8700 Spring 2014 10 Sampling Process : Signal Recovery Sampling Process : Signal Recovery Ideal low-pass filter with cut-off frequency W Recovery of the sampled signal (assume f ss 2; WT 1/(2) W ) g(t) 1 , f W HLPF ( f ) H LPF ( f ) 2W 1/(2W) GfrLPF() GfH () () f 0, f W g (t) f The impulse response is W 0 W gr (t) g (t)*hLPF (t) Ts h (t) Ts LPF g ( nT )sinc2W (t ) d sin(2Wt) 1 2W s h (t) sinc(2Wt) n LPF 2Wt T s gr (t) t g sinc2Wt n n 2W 3 1 1 1 1 3 0 2W W 2W 2W W 2W Ts Y. D. Zhang ECE-8700 Spring 2014 11 Y. D. Zhang ECE-8700 Spring 2014 12 Sampling Process : Examples Pulse-Amplitude Modulation (PAM) g(t)=cos(4000 t) W =2000 Hz, Nyquist rate 2W=4000 Hz A train of impulse sequences is good, but not practical. At sampling freq f = 6000 Hz: At sampling freq f = 1500Hz: s s PAM uses flat-top samples to substitute impulses. No aliasing: Nyquist sampling Aliasing occurred: Nyquist Pros: limit the bandwidth; can use higher power theorem is satisfied. sampling theorem not satisfied. Cons: aperture effect – discussed below PAM waveform s(t) can be considered as convolution of impulse train and rectangular window Flat-top samples, representing an analog signal Y. D. Zhang ECE-8700 Spring 2014 13 Y. D. Zhang ECE-8700 Spring 2014 14 PAM : Aperture Effect PAM : Aperture Effect Distortion in the magnitude and time If the duty cycle T/Ts is small ( T / T s 0 . 1 ), then the delay due to the use of flap-top pulse. aperture effect is insignificant. H()f T sinc()exp(f T jf T ) Otherwise, equalizer has to be applied, with magnitude (linear phase delay causes constant time response delay of T/2 without no waveform 11 f distortion) | H()f | Tsinc(f T ) sin( f T ) Org signal spectrum PAM signal spectrum Y. D. Zhang ECE-8700 Spring 2014 15 Y. D. Zhang ECE-8700 Spring 2014 16 Other Forms of Pulse Modulation PDM & PPM : Observations Pulse-Duration Modulation PDM is energy-inefficient (PDM) or Pulse-Width PPM Modulation (PWM) . For perfect rectangular pulse (which requires high Pulse-Position Modulation bandwidth), the effect of noise is negligible. (PPM) . Exhibit a threshold effect as the SNR is low Figures: PDM and PPM for the . Used in Ultra-Wideband (UWB) communications case of a sinusoidal modulating wave. (a) Modulating wave. (b) Pulse carrier. (c) PDM wave. (d) PPM wave. Y. D. Zhang ECE-8700 Spring 2014 17 Y. D. Zhang ECE-8700 Spring 2014 18 Effective Data Representation Quantization : Example 24 bits (8-bit in each color) 6 bits (2-bit in each color) How many bits you need to represent 8 levels of information (0 ~7)? Amplitude based Code based Level 6 Binary code 110 . A 75% reduction of file size in raw data format can be expected; It does not necessarily reflect that in a compressed file format. Quantization is a lossy operation. Y. D. Zhang ECE-8700 Spring 2014 19 Y. D. Zhang ECE-8700 Spring 2014 20 Quantization Quantization Consider memoryless or instantaneous quantization Quantization: Transform the continuous-amplitude m=m(nTs) process, we remove nTs for notational convenience. to discrete approximate amplitude v=v(nTs). Such a discrete approximate is adequately good in the Types of quantization: sense that any human ear or eye can detect only finite Uniform: Quantization step sizes are equal intensity differences. Non-uniform: Quantization step sizes are not equal Mid-read Mid-rise Description of a memoryless quantizer. ( (a) midtread (b) midrise Y. D. Zhang ECE-8700 Spring 2014 21 Y. D. Zhang ECE-8700 Spring 2014 22 Quantization Noise Quantization Noise Define the quantization noise to be QMVMgM() Consider uniform quantization. Assume message M is uniformly distributed in (–mmax, mmax), so M has zero mean. Assume g(.) is symmetric. Then V=g(M) also has zero mean. Therefore, Q=M–V also has zero mean. If we choose [-mmax, mmax] as the quantization range (full-load quantizer), then the step size of the quantizer is given by 2m max L where L is the total number of representation levels. Y. D. Zhang ECE-8700 Spring 2014 23 Y. D. Zhang ECE-8700 Spring 2014 24 Quantization Noise Quantization Noise Quantization error is bounded by /2 q /2 If we construct a binary code, and denote R as the bits per samples, then R RL log ( ) If the step size D is sufficiently small, it is reasonable to L 2 2 assume that Q is a uniformly distributed random variable. That is, Rewrite the mean-square value of the quantization noise as 1 , q 2 2 22 22 2 112mmmmax max max Q 22 R 12 12LL 3 3 2 fqQ () 0, otherwise Then, the output SNR (signal to quantization noise ratio) of a uniform quantizer is given by PP3 2 R Its mean-square value is given by ()SNR O 22 2 2 Q m max /21 /2 22EQ[] q 2 f () qdq qdq 2 QQ/2 /2 12 The SNR increases exponentially with the R: (SNR)O (dB) = a + 6R (the value of a varies) Y. D. Zhang ECE-8700 Spring 2014 25 Y. D. Zhang ECE-8700 Spring 2014 26 Example: Sinusoidal Modulating Signal Effect of Quantization Loading Level A 2 Let mt () A cos(2 ft ). Then, P m , mA , mc 2 max m 2 33P 222RRR3/2m max ()SNRO 22 2 2 2 (1.86)dB R mmmax max 2 LR(SNR)O 32 5 31.8 64 6 37.8 128 7 43.8 256 8 49.8 In this example, we assumed full-load quantizer, in which no quantization loss is encountered due to saturation. In general, (SNR)O is expressed (a + 6R) dB, with a The output SNR is reduced when full-load is not appropriate. depending on the waveform and the quantization load. Y. D. Zhang ECE-8700 Spring 2014 27 Y. D. Zhang ECE-8700 Spring 2014 28 Non-Uniform Quantization Non-Uniform Quantization Why: To give more protection to weak signals (2) -law A | m | 1 How: Apply compressor before quantization. , 0 | m | | v | 1 log A A 1 log( A | m |) 1 Compression laws , | m |1 (1) -law 1 log A A log( 1 | m |) Compression: at transmitter | v | log( 1 ) according to the -law or A-law Expansion: at receiver to When m 1, v m restore the signal samples to their correct relative level log( 1 | m |) | m | When m 1, v log( m ) [Compressor + Expander = Compander] Y.