Chapter 4: Sampling and Quantization A First Course in Digital Communications Ha H. Nguyen and E. Shwedyk February 2009 A First Course in Digital Communications 1/41
Chapter 4: Sampling and Quantization Introduction Though many message sources are inherently digital in nature, two of the most common message sources, audio and video, are analog, i.e., they produce continuous time signals. To make analog messages amenable for digital transmission sampling, quantization and encoding are required. Sampling : How many samples per second are needed to exactly represent the signal and how to reconstruct the analog message from the samples? Quantization : To represent the sample value by a digital symbol chosen from a finite set. What is the choice of a discrete set of amplitudes to represent the continuous range of possible amplitudes and how to measure the distortion due to quantization? Encoding : Map the quantized signal sample into a string of digital, typically binary, symbols. A First Course in Digital Communications 2/41
Chapter 4: Sampling and Quantization Ideal (or Impulse) Sampling ✁ ∞ m ( t m t = m nT δ t − nT ) ( ) ( ) ( ) s s s = −∞ n � ∞ s t = δ t − nT ( ) ( ) s ✂ ☎ n = −∞ ✄ T s is the period of the impulse train, also referred to as the sampling period . The inverse of the sampling period, f s = 1 /T s , is called the sampling frequency or sampling rate . It is intuitive that the higher the sampling rate is, the more accurate the representation of m ( t ) by m s ( t ) is. What is the minimum sampling rate for the sampled version m s ( t ) to exactly represent the original analog signal m ( t ) ? A First Course in Digital Communications 3/41
Chapter 4: Sampling and Quantization Illustration of Ideal Sampling ✞ ✟ ✆ ✝ M ( f ) M ( 0 ) ✡ ✏ ✎ ✏ ☛ ✑ ✠ ✟ ☞ ✍ ✝ ✌ ✕ ✒ ✓ ✔ ✛ ✜ ✢ ✣ ✜ ★ ✩ ✪ ✦ ✣ ✕ ✔ ✚ ✒ ✢ ✙ ✗ ✬ ✢ ✦ ✢ ✦ ✥ ✢ ✦ ✧ ★ ✩ ✪ ✦ ✭ ✢ ✦ ✘ ✫ ✫ ✕ ✜ ✤ ✣ ✔✖ ✱ ✲ ✮ ❀ ❁ ✯ ✰ ❂ ❃ ❄ ❅❅ ❆ ❇ ❈ ❉ ❊❋ ● ❄ ❇ ❍ ■ ❏ M s ( f ) M T ( 0 ) s ✵ ✴ ✶ ✿ ✻ ✺ ✻ ✿ ✯ ✾ ✶ ✽ ✾ ✶ ✽ ✼ ✶ ✽ ✶ ✽ ✲ ✷ ✸ ✹ ✰ ✳ A First Course in Digital Communications 4/41
Chapter 4: Sampling and Quantization Spectrum of the Sampled Waveform m s ( t ) = m ( t ) s ( t ) ↔ M s ( f ) = M ( f ) ∗ S ( f ) � � ∞ ∞ � � 1 = 1 M s ( f ) = M ( f ) ∗ δ ( f − nf s ) M ( f − nf s ) . T s T s n = −∞ n = −∞ � �� � S ( f ) If the bandwidth of m ( t ) is limited to W Hertz, m ( t ) can be completely recovered from m s ( t ) by an ideal lowpass filter of bandwidth W if f s ≥ 2 W . When f s < 2 W (under-sampling), the copies of M ( f ) overlap and it is not possible to recover m ( t ) by filtering ⇒ aliasing . A First Course in Digital Communications 5/41
Chapter 4: Sampling and Quantization Reconstruction of m ( t ) � � ∞ ∞ M s ( f ) = F{ m s ( t ) } = m ( nT s ) F{ δ ( t − nT s ) } = m ( nT s )exp( − j 2 πnfT s ) n = −∞ n = −∞ � ∞ M ( f ) = M s ( f ) = 1 m ( nT s )exp( − j 2 πnfT s ) , − W ≤ f ≤ W. f s f s n = −∞ � ∞ F − 1 { M ( f ) } = m ( t ) = M ( f )exp( j 2 πft )d f −∞ � W � ∞ 1 = m ( nT s )exp( − j 2 πnfT s )exp( j 2 πft )d f f s − W n = −∞ � W � ∞ 1 = m ( nT s ) exp[ j 2 πf ( t − nT s )]d f f s − W n = −∞ � n � � � ∞ ∞ m ( nT s )sin[2 πW ( t − nT s )] = = m sinc(2 Wt − n ) πf s ( t − nT s ) 2 W n = −∞ n = −∞ A First Course in Digital Communications 6/41
Chapter 4: Sampling and Quantization Sampling Theorem Theorem A signal having no frequency components above W Hertz is completely described by specifying the values of the signal at periodic time instants that are separated by at most 1 / 2 W seconds. f s ≥ 2 W is known as the Nyquist criterion , the sampling rate f s = 2 W is called the Nyquist rate and its reciprocal called the Nyquist interval . Ideal sampling is not practical ⇒ Need practical sampling methods. A First Course in Digital Communications 7/41
Chapter 4: Sampling and Quantization Bandlimited Interpolation Example of Band−limited Signal Reconstruction (Interpolation) 1 x ( t ) 0.5 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 1 x [ n ]= x ( nT s ) 0.5 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 1 x r ( t ) 0.5 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 Normalized time ( t / T s ) A First Course in Digital Communications 8/41
Chapter 4: Sampling and Quantization Natural Sampling ▲ ∞ = − m ( t m t m t h t nT ) ( ) ( ) ( ) s s n = −∞ ❑ ∞ = − p t h t nT ( ) ( ) s n = −∞ ▼ ❖ ◆ In the above, h ( t ) = 1 for 0 ≤ t ≤ τ and h ( t ) = 0 otherwise. The pulse train p ( t ) is also known as the gating waveform. Natural sampling requires only an on/off gate. A First Course in Digital Communications 9/41
Chapter 4: Sampling and Quantization Illustration of Natural Sampling ❘ ❙ P ◗ M ( f ) M ( 0 ) ❯ ❱ ❬ ❩ ❬ ❭ ❚ ❙ ❲ ❨ ◗ ❳ ❵ ❴ ❪ ❫ P ( f ) ( ) ( ) ( ) D D D ❝ 2 0 1 τ ❜ ❞ ❡ r ❦ ♦ ♣ q q ❢ ❡ ❢ ❣ ❣ ❧ ❣ ❧ ❣ ❧ ♠ ♥ ❧ s ❣ ❧ ❵ ❤ ✐ ❥ ❫❛ ① t ✇ ✉ ✈ ❺ ❻ ❼ ❽ ❾ ❿❿ ➀ ➁ ➂ ➃ ➄➅ ➆ M s ( f ) ④ ③ ⑤ ❹ ⑩ ⑨ ⑩ ❹ ✉ ❸ ⑤ ❷ ❸ ⑤ ❷ ❶ ⑤ ❷ ⑤ ❷ ② ① ⑥ ⑦ ⑧ ✈ A First Course in Digital Communications 10/41
Chapter 4: Sampling and Quantization Signal Reconstruction in Natural Sampling Write the periodic pulse train p ( t ) in a Fourier series as: � nτ � ∞ � D n = τ e − jπnτ/T s . p ( t ) = D n exp( j 2 πnf s t ) , sinc T s T s n = −∞ The sampled waveform and its Fourier transform are ∞ � m s ( t ) = m ( t ) D n exp( j 2 πnf s t ) . n = −∞ ∞ ∞ � � M s ( f ) = D n F{ m ( t )exp( j 2 πnf s t ) } = D n M ( f − nf s ) . n = −∞ n = −∞ The original signal m ( t ) can still be reconstructed using a lowpass filter as long as the Nyquist criterion is satisfied. A First Course in Digital Communications 11/41
Chapter 4: Sampling and Quantization Flat-Top Sampling Flat-top sampling is the most popular sampling method and involves two simple operations: sample and hold . ➊ ➉ ➇ ➈ ➊ ➉ ➇➌ ➈ ➉ ➋ ➌ ➍ ➎ ➌ τ ➎ ➑ h ( t ) ➐ ➒ ∞ ∞ m ( t ) ➓ m nT δ t − nT t m t = m nT h t − nT ( ) ( ) ( ) ( ) ( ) τ s s s s s n = −∞ n = −∞ ➏ ∞ s t = δ t − nT ( ) ( ) ➔ s → ➣ n = −∞ A First Course in Digital Communications 12/41
Chapter 4: Sampling and Quantization Spectrum of m s ( t ) in Flat-Top Sampling � � � ∞ m s ( t ) = m ( t ) δ ( t − nT s ) ∗ h ( t ) . n = −∞ � � � � ∞ ∞ F{ h ( t ) } = 1 M s ( f ) = F m ( t ) δ ( t − nT s ) H ( f ) M ( f − nf s ) , T s n = −∞ n = −∞ where H ( f ) = F{ h ( t ) } = τ sinc( fτ )exp( − jπfτ ) . ➠ ➫ ➭ ➯➲ ➳ ➵ ➲ ➳ ➻ ➼ ➽ ➢ ➸ ➺ M s ( f ) M ( f ) ↔ ➟ ➝ ➜ ➝ ➩ ➟ ➧ ➟ ➧ ➥ ➤ ➥ ➟ ➧ ➩ ➟ ➧ ➞ ➨ ➨ ➦ ➠ ➡ ➢ ↕ ➙ ➛ ❰ Ï ➴ ➬ Ð Ñ Ò ÓÓ Ô Õ Ö × Ø Ù Ú Ï Ö Ò × × Ñ ÓÒ Û Ñ Ö Õ Ü Ý ❒ Þ H ( f M s ( f ) ) ➾ ❮ ➱ ➮ ➱ ❮ ➹ ❒ ❒ ✃ ➘ ➘ ❐ ➘ ❐ ➘ ❐ ➘ ❐ ➚ ➶ ➴➷ ➬ − τ ➪ τ 1 1 A First Course in Digital Communications 13/41
Chapter 4: Sampling and Quantization Equalization Not possible to reconstruct m ( t ) using an lowpass filter, even when the Nyquist criterion is satisfied. The distortion due to H ( f ) can be corrected by connecting an equalizer in cascade with the lowpass reconstruction filter. Ideally, the amplitude response of the equalizer is T s T s | H eq | = | H ( f ) | = τ sinc( fτ ) á â ã ã ê ê ì â çè ç ç è í î ä å æ é ä å ë ì ê ð â è é ï ß T = H s eq τ f τ à sinc( ) m s ( t m ( t ) ) f − W W ñ ò ó A First Course in Digital Communications 14/41
Chapter 4: Sampling and Quantization Pulse Modulation In pulse modulation, some parameter of a pulse train is varied in accordance with the sample values of a message signal. Pulse-amplitude modulation (PAM): amplitudes of regularly spaced pulses are varied. PAM transmission does not improve the noise performance over baseband modulation, but allows multiplexing, i.e., sharing the same transmission media by different sources. The multiplexing advantage offered by PAM comes at the expense of a larger transmission bandwidth. Pulse-width modulation (PWM): widths of the individual pulses are varied. Pulse-position modulation (PPM): position of a pulse relative to its original time of occurrence is varied. Pulse modulation techniques are still analog modulation. For digital communications of an analog source, quantization of sampled values is needed. A First Course in Digital Communications 15/41
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