8.4 Pulse Modulation and Coding

8.4 Pulse Modulation and Coding

Introduction to Pulse Modulation

Pulse modulation techniques involve representing analog information signals using discrete pulses. Unlike continuous-wave modulation (AM, FM), pulse modulation is inherently suitable for time-division multiplexing (TDM) and digital signal processing. These techniques form the crucial bridge between analog signals and their digital representation, enabling efficient storage, transmission, and processing. This section explores the family of pulse modulation schemes, from the foundational Pulse Amplitude Modulation (PAM) to the ubiquitous Pulse Code Modulation (PCM) and its more efficient derivatives like DPCM and Delta Modulation, along with their inherent noise and bandwidth considerations.

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1. Pulse Amplitude Modulation (PAM)

1.1 Basic Principle

1. Definition: The amplitudes of regularly spaced, uniform-width pulses are varied in proportion to the instantaneous sample values of the continuous message signal $m(t)$.

2. Generation Process:

   • Sampling: The message signal $m(t)$ is sampled at a rate $f_s \ge 2f_m$ (Nyquist rate).

   • Pulse Shaping: Each sample value modulates the amplitude of a carrier pulse sequence.

3. Types:

   • Natural Sampling (Gating): The amplitude of each pulse follows the natural shape of the message signal during the pulse duration.

   • Flat-Top Sampling: The amplitude of each pulse is held constant at the sample value for the pulse duration. This is more common in practice and is generated using a Sample-and-Hold (S/H) circuit.

1.2 Mathematical Representation

For flat-top PAM with pulse shape $p(t)$: $s_{\text{PAM}}(t) = \sum_{n=-\infty}^{\infty} m(nT_s) \cdot p(t - nT_s)$ where $T_s = 1/f_s$ is the sampling interval, and $p(t)$ is typically a rectangular pulse.

1.3 Bandwidth Considerations

1. Key Insight: A PAM signal's spectrum is the original message spectrum $M(f)$ repeated at integer multiples of the sampling frequency $f_s$.

2. Spectrum:

   • For ideal instantaneous sampling, the spectrum is: $S_{\text{PAM}}(f) = f_s \sum_{k=-\infty}^{\infty} M(f - kf_s)$

   • For flat-top sampling using rectangular pulses of width $\tau$, the spectrum is multiplied by a sinc envelope: $S_{\text{Flat-top}}(f) = \frac{\tau}{T_s} \text{sinc}(f\tau) \sum_{k=-\infty}^{\infty} M(f - kf_s)$ where $\text{sinc}(x) = \sin(\pi x) / (\pi x)$.

3. Bandwidth Requirement: To reconstruct the original signal without aliasing:

   • The sampling rate must satisfy: $f_s > 2f_m$.

   • The required channel bandwidth $B$ must be at least $f_m$ (the bandwidth of the original signal). In practice, due to the sinc roll-off and the need for guard bands, $B \approx \frac{1}{\tau}$, where $\tau$ is the pulse width. Narrower pulses require larger bandwidth.

4. Demodulation: Simple low-pass filtering (with cutoff $f_m$) can reconstruct the original signal from natural PAM. For flat-top PAM, an equalizer (with response $1/\text{sinc}(f\tau)$) is needed before the LPF to correct for the aperture effect.

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2. Pulse Code Modulation (PCM)

PCM is the complete process of converting an analog signal into a digital bit stream. PAM is its first step.

2.1 The PCM Process Block Diagram

Analog Input → Sampler → Quantizer → Encoder → Digital Bit Stream (PCM)

2.2 Encoders and Decoders

1. Encoder: Performs the three steps: Sampling, Quantization, and Encoding. It outputs a serial binary stream.

   • Types: Flash ADC (parallel comparator type) is common for high-speed applications.

2. Decoder: Performs the inverse operations at the receiver.

   • Digital-to-Analog Converter (DAC): Converts the digital PCM words back into quantized PAM pulses (a staircase approximation).

   • Reconstruction (Smoothing) Low-Pass Filter: Smoothens the staircase output to recover the original analog signal, attenuating high-frequency sampling components.

2.3 Quantization and Quantization Error

1. Quantization: The process of mapping the continuous range of sample amplitudes into a finite set of discrete quantization levels.

2. Quantizer Characteristics:

   • Step Size ($\Delta$): The voltage difference between two adjacent quantization levels. $\Delta = \frac{V_{pp}}{L}$, where $V_{pp}$ is the peak-to-peak input range and $L$ is the number of levels.

   • Mid-Rise vs. Mid-Tread Quantizers:

     • Mid-Rise: Has a level at zero. Used for signals with no DC offset.

     • Mid-Tread: Has zero as a decision level, not a reconstruction level. Better for small signals as they are quantized to zero.

3. Quantization Error (Noise):

   • Definition: The difference between the original sample value and the quantized value. $q = m(nT_s) - m_q(nT_s)$

   • Model: For a uniform quantizer and a sufficiently busy input signal, the quantization error can be modeled as a uniformly distributed random variable over the range $[-\Delta/2, +\Delta/2]$.

   • Mean-Square Quantization Error (Noise Power): $\sigma_q^2 = \frac{1}{\Delta} \int_{-\Delta/2}^{\Delta/2} q^2 \, dq = \frac{\Delta^2}{12}$

2.4 Signal-to-Quantization-Noise Ratio (SQNR)

1. Definition: The ratio of the signal power to the quantization noise power, a key metric of PCM quality.

2. For a Sinusoidal Test Signal:

   • Let the sinusoid amplitude be $A$, fitting within the quantizer range.

   • Signal Power: $P_s = A^2/2$.

   • For an L-level uniform quantizer, $\Delta = 2A / L$.

   • Quantization Noise Power: $P_q = \Delta^2 / 12 = A^2 / (3L^2)$.

   • SQNR: $\text{SQNR} = \frac{P_s}{P_q} = \frac{(A^2/2)}{(A^2/(3L^2))} = \frac{3}{2} L^2$

3. In Decibels (dB) and in terms of bits (n): Since $L = 2^n$, where $n$ is the number of bits per sample, $\text{SQNR (dB)} = 10 \log_{10}\left(\frac{3}{2} L^2\right) = 10 \log_{10}(1.5) + 20 \log_{10}(L)$ $\text{SQNR (dB)} \approx 1.76 + 6.02n$ Interpretation: Each additional bit improves SQNR by approximately 6 dB.

2.5 Companding (Compressor/Expander)

1. Problem: The SQNR formula shows that for a fixed n, weak signals suffer from a much lower SQNR than strong signals because $\Delta$ is fixed. This is inefficient for signals like speech with a wide dynamic range.

2. Solution: Non-Uniform Quantization via Companding.

3. Principle:

   • At Transmitter (Compression): Pass the signal through a compressor – a nonlinear amplifier with a logarithmic-like characteristic (e.g., μ-law, A-law). It amplifies weak signals more than strong signals.

   • Uniform Quantization: The compressed signal is then quantized using a uniform quantizer.

   • At Receiver (Expansion): The quantized values are passed through an expander, which has the inverse characteristic of the compressor. The combination is called a compander.

4. Effect: Effectively provides a small step size ($\Delta$) for small signals and a large step size for large signals. This yields a more constant SQNR across the input signal's dynamic range.

5. Standards:

   • μ-law: Used in North America and Japan. $y = \frac{\log(1 + \mu |x|)}{\log(1+\mu)} \cdot \text{sgn}(x)$

   • A-law: Used in Europe and international systems. $y = \begin{cases} \frac{A|x|}{1+\ln(A)}, & |x| \le \frac{1}{A} \\ \frac{1+\ln(A|x|)}{1+\ln(A)}, & \frac{1}{A} \le |x| \le 1 \end{cases}$

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3. Multiplexing in PCM Systems

PCM is naturally suited for Time Division Multiplexing (TDM).

1. Principle: Multiple PCM streams (from different message sources) are interleaved in the time domain to share a single high-speed digital transmission link.

2. Frame Structure:

   • A frame is a time interval containing one PCM word (sample) from each of the $N$ input channels.

   • Frame duration $T_F = 1/f_s$ (the sampling interval for any one channel).

   • Each channel's time slot within a frame has duration $T_s / N$.

3. Bit Rate of TDM-PCM System: $R_b = N \times f_s \times n$ where $N$ is the number of channels, $f_s$ is the sampling rate per channel, and $n$ is the number of bits per sample.

4. Example - The T1 Digital Carrier System (North America):

   • $N = 24$ voice channels.

   • $f_s = 8$ kHz.

   • $n = 8$ bits/sample (using companding).

   • Frame length = 24×8 = 192 bits + 1 framing bit = 193 bits/frame.

   • Frame rate = 8000 frames/sec.

   • Line Bit Rate: $R_b = 193 \times 8000 = 1.544$ Mbps.

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4. Differential PCM (DPCM) and Noise

4.1 Basic Principle

1. Motivation: In many signals (like speech, video), there is high correlation between successive samples. The difference between consecutive samples has lower variance than the signal itself.

2. Core Idea: Instead of quantizing and transmitting the sample value $m(nT_s)$ itself, transmit the quantized version of the prediction error $e(n)$. $e(n) = m(n) - \hat{m}(n)$ where $\hat{m}(n)$ is a predicted value of $m(n)$, typically the previously reconstructed sample.

3. System Operation:

   • Transmitter: Quantizes the difference $e(n)$ and sends it. It also locally reconstructs $\hat{m}(n)$ using the same predictor as the receiver.

   • Receiver: Uses the received quantized difference to update its predictor and reconstructs $\hat{m}(n)$.

4.2 Advantages over Standard PCM

• Bit Rate Reduction: For the same quantization error (quality), DPCM can use fewer bits per sample ($n$) than standard PCM because the variance of $e(n)$ is smaller.

• Alternatively, for the same bit rate (same $n$), DPCM provides a higher SQNR than PCM.

4.3 Noise in DPCM

1. Quantization Noise: Present as in PCM, but affects the difference signal.

2. Slope Overload Noise (a form of distortion): Occurs when the input signal changes too rapidly (high slope). The predictor (often just a previous sample) cannot track it, causing large prediction errors that exceed the quantizer's maximum range. This leads to gross distortion.

3. Granular Noise: Occurs when the input signal is nearly constant. The quantized difference signal oscillates between ±Δ/2, causing a sawtooth pattern in the reconstructed signal.

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5. Delta Modulation (DM)

5.1 Basic Principle

Delta Modulation is a 1-bit DPCM system ($n=1$). It is the simplest form of differential coding.

1. Operation:

   • The quantizer has only two levels (+Δ and -Δ), representing the sign of the difference.

   • The predictor is a simple integrator (accumulator).

   • At each step, the encoder transmits a single bit: '1' if the signal increased, '0' if it decreased.

   • The receiver uses an identical integrator to reconstruct a staircase approximation of the input.

5.2 Mathematical/Logical Operation

• Let $b(n)$ be the output bit (1 or 0).

• Transmitter: $b(n) = \begin{cases} 1 & \text{if } m(n) > \hat{m}(n-1) \\ 0 & \text{if } m(n) < \hat{m}(n-1) \end{cases}$ and updates: $\hat{m}(n) = \hat{m}(n-1) + \Delta \cdot (2b(n)-1)$

• Receiver: Uses the same update equation with the received bit stream.

5.3 Noise in Delta Modulation

DM faces a critical trade-off governed by the step size ($\Delta$) and sampling rate ($f_s$).

1. Slope Overload Distortion:

   • Cause: Occurs when the input signal slope exceeds the maximum tracking slope of the DM system, which is $\Delta / T_s = \Delta f_s$. $\left| \frac{dm(t)}{dt} \right|_{max} > \Delta f_s$

   • Effect: The staircase approximation cannot keep up with a rapidly rising or falling input, causing large, distorted error.

2. Granular Noise:

   • Cause: Occurs when the input signal is relatively flat. The staircase approximation oscillates around the true signal with an amplitude of ~Δ/2, producing a sawtooth pattern.

   • Effect: A constant, unwanted "hiss" or chatter in the reconstructed signal.

5.4 Adaptive Delta Modulation (ADM)

1. Motivation: To resolve the DM trade-off. A fixed Δ is too small to handle slope overload for fast signals and too large for granular noise in slow signals.

2. Principle: The step size ($\Delta$) is varied adaptively based on the recent bit pattern.

   • If consecutive bits are the same (e.g., 1, 1, 1): The signal is likely changing rapidly. The step size is increased (e.g., multiplied by a factor >1) to avoid slope overload.

   • If bits alternate (e.g., 1, 0, 1): The signal is likely near a constant value. The step size is decreased (e.g., multiplied by a factor <1) to reduce granular noise.

3. Algorithm: Various algorithms exist (e.g., Song algorithm, CVSD - Continuously Variable Slope Delta modulation).

4. Advantage: Provides much better dynamic range and fidelity than basic DM for the same bit rate.

5.5 Sigma-Delta Modulation (ΣΔ)

1. Also Known As: Delta-Sigma Modulation.

2. Key Innovation: Places an integrator before the delta modulator (in the signal path), and a differentiator after the delta demodulator (in the receiver path).

   • The integrator in the transmitter acts as a pre-emphasis filter, emphasizing low frequencies.

   • The 1-bit quantizer and feedback operate on the integrated signal.

   • The differentiator in the receiver acts as a de-emphasis filter, recovering the original signal while also acting as a noise-shaping filter.

3. Noise Shaping: The quantization error in ΣΔ modulation is high-pass filtered. Most of the quantization noise power is pushed to high frequencies, out of the signal band.

4. Oversampling: ΣΔ modulators operate at a sampling rate $f_s$ much higher than the Nyquist rate (often 64x or 128x). Combined with noise shaping and digital filtering, this allows the use of a 1-bit quantizer to achieve very high resolution (e.g., 16-24 bits effective) for audio signals.

5. Primary Application: High-resolution Analog-to-Digital Converters (ADCs) and Digital-to-Analog Converters (DACs) used in digital audio (CD players, professional audio equipment).

Conclusion: Pulse modulation techniques provide the essential link between the analog and digital worlds. PCM, with its robust process of sampling, quantization, and encoding, is the foundation of digital telephony and audio. Understanding its limitations in terms of SQNR and dynamic range leads to the use of companding. The desire for greater efficiency leads to DPCM, while the need for extreme simplicity drives the development of DM and its adaptive variant, ADM. Finally, Sigma-Delta modulation represents a sophisticated blend of oversampling and feedback that enables high-fidelity conversion with simple 1-bit circuitry, illustrating the continuous evolution of pulse modulation and coding principles.