ADC Unit-IV

Analog and Digital Communications

UNIT - IV: Pulse Modulation

Contents

• Pulse Analog Modulation

  • Types of Pulse analog modulation

    • PAM (Pulse Amplitude Modulation)

    • PWM (Pulse Width Modulation)

    • PPM (Pulse Position Modulation)

  • Comparison of FDM and TDM

• Pulse Digital Modulation

  • PCM Generation and Reconstruction

  • Quantization Noise

  • Non-Uniform Quantization and Companding

  • DPCM (Differential Pulse Code Modulation)

  • DM (Delta Modulation)

  • Adaptive DM (Adaptive Delta Modulation)

  • Noise in PCM and DM

• Applications: Design of E1 and T1 digital-carrier systems

Types of Modulation

Mr. CH Sathyanarayana's lecture classifies modulation techniques into analog and digital modulation.

• Analog Modulation

  • Continuous Wave

    • Amplitude Modulation (AM)

    • Angle Modulation

      • Frequency Modulation (FM)

      • Phase Modulation (PM)

  • Pulse Modulation

    • Pulse Analog

      • PAM

      • PWM

      • PPM

    • Pulse Digital

      • PCM

      • DPCM

      • DM

      • ADM

• Digital Modulation

  • ASK (Amplitude Shift Keying)

  • FSK (Frequency Shift Keying)

  • PSK (Phase Shift Keying)

  • QAM (Quadrature Amplitude Modulation)

Pulse Analog Modulation Techniques

Introduction

Following continuous wave modulation, the lecture transitions to pulse modulation, focusing on analog pulse modulation techniques.

• Pulse Amplitude Modulation (PAM)

• Pulse Width Modulation (PWM)

• Pulse Position Modulation (PPM)

Pulse Amplitude Modulation (PAM)

In PAM, the amplitude of the pulse carrier varies proportionally to the instantaneous amplitude of the message signal.

The pulse amplitude modulated signal mirrors the amplitude of the original signal.

In natural PAM, a signal sampled at the Nyquist rate can be reconstructed by passing it through a low-pass filter (LPF) with an exact cutoff frequency.

Pulse Width Modulation (PWM)

Refer to Electronic Communication Systems by George Kennedy for detailed information on PWM.

• PWM Signal

  • Carrier Sampling Pulses

  • Modulating Signal

Pulse Position Modulation (PPM)

Refer to Electronic Communication Systems by George Kennedy for detailed information on PPM.

• Generation of Pulse Position Modulation

  • Signal

  • PWM

  • Differentiated

  • Clipped (PPM)

Time Division Multiplexing (TDM)

TDM is a technique for transmitting several analog message signals over a single communication channel by dividing the time frame into slots, one for each signal.

Four input signals, each band-limited to $f_x$ by input low-pass filters, are sequentially sampled at the transmitter using a rotary switch (commutator).

The commutator makes $f_s$ revolutions per second, extracting one sample from each input during each revolution.

The output of the switch is a PAM waveform containing periodically interfaced samples of the input signals.

TDM: Example

Suppose we have four sources, each creating 250 characters per second. If the interleaved unit is a character and 1 synchronizing bit is added to each frame:

1. The data rate of each source is 2000 bps = 2 Kbps.

2. The duration of a character is $1/250$ s, or 4 ms.

3. The link needs to send 250 frames per second.

4. The duration of each frame is $1/250$ s, or 4 ms.

5. Each frame is $4 \times 8 + 1 = 33$ bits.

6. The data rate of the link is $250 \times 33$, or 8250 bps.

Statistical TDM

Addressing is required in Statistical TDM. The slot size, i.e., the ratio of the data size to the address size must be reasonable to make transmission efficient.

No synchronization bit is needed, meaning no need for frame-level sync.

Bandwidth is normally less than the sum of the capacities of each channel.

Comparison of FDM and TDM

Elements of a Communication System

The basic elements are Source, Transmitter, Channel, Receiver, and Destination.

Model of Digital Communication System

The model includes the following components:

• Digital message source

• Analog message source

• Analog-to-digital converter

• Source encoder

• Channel encoder

• Digital modulator (transmitter side)

• Channel (with noise and interference)

• Digital demodulator (receiver side)

• Channel decoder

• Source decoder

• Digital-to-analog converter

• Analog message output signal

• Digital message output signal

Source Encoder

The aim of the source coding is to remove the redundancy present in the message so that bandwidth required for transmission is minimized.

It converts the input (symbol sequences) into binary sequences of 0’s and 1’s by assigning code words to the input symbol sequences.

The important parameters of a source encoder are codeword, entropy, and efficiency of the coder.

Source Decoder

Source decoder converts the binary output into symbol sequences.

Channel Encoder/Decoder

1. The channel encoder adds redundant bits to the message.

2. The channel decoder detects and corrects the errors present in the received signal with the help of added redundant bits.

3. In general, channel encoder divides the input message bits into blocks of k bits and replaces each k-bit message block by an n-bit code word by introducing $(n-k)$ check bits to each message block.

Modulator

Converts the input bit stream into an electrical waveform suitable for transmission over the communication channel.

It should be designed to minimize the effects of channel noise and to match the frequency spectrum of the transmitted signal with channel characteristics.

Demodulator

The extraction of the message from the information-bearing waveform produced by the modulation is accomplished by the demodulator. The output of the demodulator is a bit stream.

Channel

Provides the electrical connection between the source and destination.

Different channels include Pair of wires, Coaxial cable, Optical fiber, Radio channel, Satellite channel, or a combination of any of these.

The communication channels have finite bandwidth, non-ideal frequency response, and the signal often suffers from amplitude and phase distortion as it travels over the channel.

The signal power decreases (with distance) due to the attenuation of the channel.

The signal is corrupted by unwanted, unpredictable electrical signals referred to as noise.

The important parameters of the channel are bandwidth, amplitude and phase response, and the statistical properties of noise.

Advantages and Disadvantages of Digital Communication System

Advantages

• More immune to noise and interference

• Makes communication reliable with added error detection and correction techniques

• Provides added security

• Easy Multiplexing

• Simpler, easier, and cheaper compared to analog systems (advancements in IC technology made it possible)

Disadvantages

• Requires more transmission bandwidth.

• Needs synchronization in case of synchronous operation.

Sampling Theorem

Sampling Theorem: “A band-limited signal can be reconstructed from its samples if the sampling rate is at least equal to twice the maximum frequency component present in it.”

Let's say the maximum frequency component of $g(t)$ is $f<em>m$. To recover the signal $g(t)$ from its samples, it has to be sampled at a rate $f</em>s ≥ 2f_m$.

The minimum required sampling rate $f<em>s = 2f</em>m$ is called the Nyquist rate.

Proof

Let $g(t)$ be a band-limited signal whose bandwidth is $f<em>m$ ($\omega</em>m = 2\pi f_m$).

$\delta<em>T(t)$ is the sampling signal with fs = 1/T > 2f_m.

Let $g<em>s(t)$ be the sampled signal. Its Fourier Transform $G</em>s(\omega)$ is given by

$F(g<em>s(t)) = G</em>s(\omega) = F[g(t)\delta<em>T(t)] = F[g(t) \Sigma</em>{n=-\infty}^{+\infty} \delta(t - nT)]$

$= \frac{1}{T} [G(\omega) * \omega<em>0 \Sigma</em>{n=-\infty}^{+\infty} \delta(\omega - n\omega<em>0)] = \frac{2\pi}{T} \Sigma</em>{n=-\infty}^{+\infty} G(\omega) * \delta(\omega - n\omega<em>0) = \frac{1}{T} \Sigma</em>{n=-\infty}^{+\infty} G(\omega - n\omega_0)$

If $\omega<em>s = 2\omega</em>m$, i.e., $T = 1/(2f<em>m)$. Therefore, $G</em>s(\omega)$ is given by

$G<em>s(\omega) = \frac{1}{T} \Sigma G(\omega - n\omega</em>m)$

To recover the original signal $G(\omega)$, filter with a Gate function, $H<em>{2\omega</em>m}(\omega)$ of width $2 \omega_m$ and scale it by $T$.

Aliasing

Aliasing is a phenomenon where the high-frequency components of the sampled signal interfere with each other because of inadequate sampling \omegas < 2\omegam.

Aliasing leads to distortion in the recovered signal. This is the reason why the sampling frequency should be at least twice the bandwidth of the signal.

Oversampling

In practice, signals are oversampled, where $f_s$ is significantly higher than the Nyquist rate to avoid aliasing.

Different Sampling Methods

• Ideal sampling

• Natural sampling

• Flat-top sampling

Nyquist Sampling Rate

• Low-pass signal: The Nyquist rate is $2 \times f_{max}$.

• Bandpass signal: The Nyquist rate is $2 \times f_{max}$.

Pulse Code Modulation (PCM)

Three processes are involved:

• The analog signal is sampled.

• The sampled signal is quantized.

• The quantized values are encoded as streams of bits.

Components of PCM Encoder

1. Sampling: PAM (Pulse Amplitude Modulation). According to the Nyquist theorem, the sampling rate must be at least 2 times the highest frequency contained in the signal.

PCM Steps

1. Consider the analog Signal $x(t)$.

2. The signal is first sampled.

3. Assign the Closest Level.

4. Each quantization level corresponds to a unique combination of bits. The analog signal is transmitted/stored as a stream of bits and reconstructed when required.

Quantized Signal

$x^{\sim}(t)$ represents the Quantized Signal.

It is apparent that the quantized signal is not exactly the same as the original analog signal, and there is a fair degree of quantization error here. However, as the number of quantization levels is increased, the quantization error is reduced, and the quantized signal gets closer and closer to the original signal.

PCM Transmitter - Receiver

In the PCM generator, the signal is first passed through a sampler which is sampled at a rate of $f<em>s$, where $f</em>s ≥ 2f_m$.

The output of the sampler $x(kT<em>s)$, which is discrete in time, is fed to a q-level quantizer. The quantizer compares the input samples $x(kT</em>s)$ with its fixed quantization levels. It assigns any one of the quantization levels to $x(kT<em>s)$ that results in minimum distortion or error. This error is called quantization error. The quantized signal $x</em>q(kT_s)$ is given to the binary encoder. The encoder converts each quantized sample to an n-bit binary word.

The receiver starts by reshaping the received pulses, removes the noise, and then converts the binary bits to analog. The received samples are then filtered by a low-pass filter; the cut-off frequency is at $f<em>c$, and $f</em>c = f<em>m$ where $f</em>m$ is the highest frequency component in the original signal.

PCM Characteristics

With PCM, the analog signal is sampled and converted to a serial n-bit binary code (for each sample). Each code has the same number of bits and requires the same length of time for transmission.

PCM is by far the most prevalent form of pulse modulation.

The term pulse code modulation is somewhat of a misnomer, as it is not really a type of modulation but rather a form of digitally coding analog signals.

PCM is the preferred method of communications within the PSTN (public switched telephone network).

Quantization

Quantization is the process of converting an infinite number of possibilities to a finite number of conditions.

PCM requires quantization to convert the analog signals containing an infinite number of amplitude possibilities to a PCM code with a limited number of combinations.

The smaller the magnitude of a quantum (step size), the better the resolution and the more accurately the quantized signal will resemble the original analog sample.

The likelihood of a sample voltage being equal to one of the possible quantization levels is remote. Therefore, each sample voltage is rounded off (quantized) to the closest available level and then converted to its corresponding PCM code.

The rounded-off error is called the quantization error ($\epsilon$).

The quantizer is known as the “mid-riser” type. For such a quantizer, slightly positive and slightly negative values of the input signal will have different levels at the output. This may be a problem when the speech signal is not present but a small noise is present at the input of the quantizer.

To avoid such a random fluctuation at the output of the quantizer, the “mid-tread” type uniform quantizer may be used.

Transmission Bandwidth of PCM

• Bit rate = Bits per sample * sampling rate

• Bandwidth required is half of the bit rate.

Calculation of SQNR in PCM

In PCM, the effects of transmission error can be ignored, but there is an effect of quantization error on signal quality.

There are so many ways to calculate SQNR for PCM, considering different conditions.

Calculation of Quantization Noise Power ($N_Q$)

Errors are introduced into the signal because of the quantization process. This error is called “quantization error” and is expressed as $\epsilon = x<em>q(kT</em>s) - x(kT_s)$

Let an input signal $x(t)$ have an amplitude in the range of $x<em>{max}$ to $-x</em>{max}$. Then the total amplitude range is:

Total amplitude = $x<em>{max} – (–x</em>{max}) = 2x_{max}$

If the amplitude range is divided into ‘q’ levels of the quantizer, the step size $\Delta$ is

If $\Delta$ is small it can be assumed that the quantization error is uniformly distributed. The quantization noise is uniformly distributed in the interval $[-\Delta/2, \Delta/2]$.

The uniform distribution of quantization error states

$f_{\epsilon}(\epsilon) = 1/\Delta$

The noise is the mean square value of noise voltage, since noise is defined by the random variable "$\epsilon$ " and $f(\epsilon)$, its mean square value is given by

$V<em>{noise}^2 = \int</em>{-\Delta/2}^{\Delta/2} \epsilon^2 f_{\epsilon}(\epsilon) d\epsilon$

Substitute the value of $f_{\epsilon}(\epsilon) = 1/\Delta$ in the equation:

$V<em>{noise}^2 = \int</em>{-\Delta/2}^{\Delta/2} \epsilon^2 (1/\Delta) d\epsilon = \frac{1}{\Delta} [\frac{\epsilon^3}{3}]_{-\Delta/2}^{\Delta/2} = [\frac{ (\Delta/2)^3}{3\Delta} - \frac{ (-\Delta/2)^3}{3\Delta}] = \frac{\Delta^2}{12}$

Calculation of Signal Power ($S_i$)

After getting an estimate of quantization noise power as above, we now have to find the signal power. In general, the signal power can be assessed if the signal statistics (such as the amplitude distribution probability) is known. The power associated with $x(t)$ can be expressed as

$S<em>i = \overline{x^2(t)} = \int</em>{-V}^{+V} x^2(t) p(x) dx$

Where $p(x)$ is the pdf of $x(t)$. In absence of any specific amplitude distribution it is common to assume that the amplitude of signal $x(t)$ is uniformly distributed between $+V$. In this case, it is easy to see that

$S<em>i = \overline{x^2(t)} = \int</em>{-V}^{+V} x^2(t) \frac{1}{2V} dx = (\frac{1}{2V}) [\frac{x^3}{3}]_{-V}^{+V} = \frac{V^2}{3}$

The above equation is $S_i = \frac{V^2}{3}$.

Now the SNR can be expressed as

$\frac{S<em>i}{N</em>o} = \frac{V^2/3}{\Delta^2/12} = \frac{V^2}{(\frac{2V}{q})^2 / 12} = \frac{3q^2}{12} = \frac{q^2}{4}$

It may be noted from the above expression that this ratio can be increased by increasing the number of quantizer levels $q$.

Also note that $S<em>i$ is the power of $x(t)$ at the input of the sampler and hence, may not represent the SQNR at the output of the low pass filter in the PCM decoder. However, for large N, small $\delta$, and ideal and smooth filtering (e.g., Nyquist filtering) at the PCM decoder, the power $S</em>o$ of the desired signal at the output of PCM decoder can be assumed to be almost the same as $S<em>i$, i.e., $S</em>o \approx S_i$.

With this justification, the SQNR at the output of a PCM codec can be expressed as,

$SQNR = \frac{S<em>o}{N</em>o} = 10 \log<em>{10} \frac{S</em>o}{N<em>o} = 10 \log</em>{10} (3 q^2) = 6.02N dB$

A few observations:

• Note that if the actual signal excursion range is less than $\pm V$, \frac{S}{N} < 6.02N dB.

• If one quantized sample is represented by 8 bits after encoding, i.e., $N = 8$, $SQNR = 48 dB$.

• If the amplitude distribution of $x(t)$ is not uniform, then the above expression may not be applicable.

Non-uniform Quantization

In a linear or uniform quantizer, the quantization error in the k-th sample is $e<em>k = |x(kT</em>s) – x<em>q(kT</em>s)|$ and the maximum error magnitude in a quantized sample is $\pm \Delta/2$.

So, if in $x(t)$, small amplitudes are more probable in the input signal than amplitudes closer to ‘$\pm V$’, then the quantization noise of such an input signal will be significant compared to the power of $x(t)$. This implies that SQNR of usually low signals will be poor and unacceptable.

In a practical PCM codec, it is often desired to design the quantizer such that the SQNR is almost independent of the amplitude distribution of the analog input signal $x(t)$.

We know that if the full range voltage is 16v, then the max quantization error will be 1v. For the low signal amplitudes like 2v, 3v etc., the max quantization error of 1v is quite high. But for large signal amplitudes near 15v, 16v etc., it can be considered small.

This is achieved by using a non-uniform quantizer. In non-uniform quantization, the step size is not fixed, and it varies as per input signal amplitude. From the figure, it can be observed that $\delta$ is small for low input signal levels, whereas it is higher for high input levels. Therefore, SQNR will be improved for low signal levels while keeping the SQNR almost the same throughout the range of the input signal.

Normally, we don’t know how the signal level varies in advance. Therefore, non-uniform quantization becomes difficult to implement.

So, a non-uniform quantizer can be considered equivalent to an amplitude pre-distortion process [denoted by $y = c(x)$ in fig.] followed by a uniform quantizer with a fixed step size ‘$\delta$’.

Mathematically, $c(x)$ should be a monotonically increasing function of ‘$x$’ with odd symmetry shown in the figure. The monotonic property ensures that $c(x) \times c^{-1}(x) = 1$.

Note that the operation of $c^{-1}(x)$ is necessary in the PCM decoder to get back the original signal undistorted. The range ‘$\pm V$’ of $x(t)$ further implies the following: $c(x) = +V$, for $x = +V$, $= 0$, for $x = 0$, $= -V$, for $x = -V$

From the figure, it can be observed that the signal is amplified for low signal levels and attenuated for high signal levels.

After this process, uniform quantization is used. At the receiver, a reverse process is done, that is, the signal is attenuated for low signal levels and amplified for high signal levels.

Thus, the compression of the signal at the transmitter and expansion at the receiver are combinedly called companding.

There are two popular standards for non-linear quantization known as:

• The $\mu$-law companding

• The A – law companding

The $\mu$-law has been popular in the US, Japan, Canada, and a few other countries, while the A-law is largely followed in Europe and most other countries, including India, adopting ITU-T standards.

$\mu$-law companding

The compression function $z(x)$ for $\mu$-law companding is

$c(x) = V \frac{\ln(1 + \mu |x|/V)}{\ln(1 + \mu)} \text{ for } -V \le x \le V$

$\mu$ is a constant here. It ranges from 0 to 255, and $\mu = 0$ corresponds to linear quantization. The typical value of $\mu$ is 100.

A-law companding

The compression function $z(x)$ for A-law companding is

$c(x) = \begin{cases} \frac{A|x|}{1 + \ln A} &amp; |x| \le \frac{V}{A} \ \frac{V(1 + \ln(A|x|/V))}{1 + \ln A} &amp; \frac{V}{A} \le |x| \le V \end{cases}$

'$A$' is a constant here, and the typical value used in practical systems is 87.5.

As an approximately logarithmic compression function is used for linear quantization, a PCM with a non-uniform quantization scheme is also referred to as “Log PCM” or “Logarithmic PCM” scheme.

Differential PCM

In a typical PCM-encoded speech waveform, there are often successive samples taken in which there is little difference between the amplitudes of the two samples.

This necessitates transmitting several identical PCM codes, which is redundant.

Differential pulse code modulation (DPCM) is designed specifically to take advantage of the sample-to-sample redundancies in typical speech waveforms.

With DPCM, the difference in the amplitude of two successive samples is transmitted rather than the actual sample. Because the range of sample differences is typically less than the range of individual samples, fewer bits are required for DPCM than conventional PCM.

The standard sampling rate for PCM of telephone grade band-limited speech signal is $f_s = 8$ Kilo samples per sec with a sampling interval of 125 $\mu$ sec. Samples of such speech signal are usually correlated as the amplitude of the speech signal does not change much within 125 $\mu$ sec.

So, if the difference between samples is quantized instead of the sample itself, then the number of quantization levels can be decreased, and thereby the number of bits required to encode. This type of digital pulse modulation scheme, in which differential quantization is used, is called DPCM.

It works on the principle of prediction. The value of the present sample is predicted from the past samples. Prediction may not be exact, but it is very close to the actual sample value.

$e(nT<em>s) = x(nT</em>s) \text{ - } x(nT_s)$

The quantizer output can be written as,

$e<em>q(nT</em>s) = e(nT<em>s) + q(nT</em>s)$

Here $q(nT_s)$ is the quantization error

$x<em>q(nT</em>s) = x(nT<em>s) + e</em>q(nT_s)$

Equation 1 is written as,

$x(nT<em>s) - x(nT</em>s) = e(nT_s)$

The value of $e(nT<em>s) + x(nT</em>s) = x(nT<em>s) + q(nT</em>s)$

$x<em>q(nT</em>s) = x(nT<em>s) x(nT</em>s) q(nT<em>s) = x(nT</em>s) + q(nT_s)$

The decoder first reconstructs the quantized output from the incoming binary signal. The prediction filter output and quantized output are summed up to give the quantized version of the original signal. Thus, the signal at the receiver differs from the actual signal by the quantization error.

Advantages

• Improved SQNR

• Reduced Bandwidth requirement

Disadvantages

• Complexity

DELTA MODULATION (DM)

Delta modulation is a single-bit PCM. With conventional PCM, each code is a binary representation of both the sign and the magnitude of a particular sample. Therefore, multiple-bit codes are required to represent the many values.

With DM, rather than transmitting a coded representation of the sample, only a single bit is transmitted.

The algorithm is quite simple. If the current sample is smaller than the previous sample, a logic 0 is transmitted. If the current sample is larger than the previous sample, a logic 1 is transmitted.

The input analog signal is sampled and converted to a PAM signal, which is compared with the output of the DAC.

The output of the DAC is a voltage equal to the regenerated magnitude of the previous sample, which was stored in the up-down counter as a binary number.

The up/down counter is incremented or decremented depending on whether the previous sample is larger or smaller than the current sample.

The up/down counter is clocked at a rate equal to the sample rate.

Initially, the up/down counter is zeroed, and the DAC is outputting 0 V.

The output of the comparator is a logic 1 condition (+V), indicating that the current sample is larger in amplitude than the previous sample. On the next clock pulse, the up/down counter is incremented to a count of 1.

Limitations/Problems of DM

• Slope overload distortion - When the analog input signal changes at a faster rate (slope of the analog signal is greater) than the DAC can track, a kind of distortion called “slope overload distortion” occurs.

  • Increasing the clock frequency reduces the probability of slope overload occurring.

  • Another way to prevent slope overload is to increase the step size.

• Granular noise - When the analog input signal has a relatively constant amplitude, the reconstructed signal has variations that were not present in the original signal. This is called granular noise.

  • It is analogous to quantization noise in conventional PCM.

  • It can be reduced by decreasing the step size.

Therefore, to reduce granular noise, a small resolution (step size) is needed, and to reduce the possibility of slope overload occurring, a large resolution (step size) is required. Obviously, a compromise is necessary.

Granular noise is more prevalent in analog signals that have gradual slopes and whose amplitudes vary only a small amount. Slope overload is more prevalent in analog signals that have steep slopes or whose amplitudes vary rapidly.

SQNR of DM

To obtain signal power, we have derived that slope overload distortion will not occur if

$A \leq \frac{\delta}{2\pi f T_s}$

Here:

• A is the peak amplitude of the sinusoidal signal

• $\delta$ is the step size

• f is the signal frequency and

• Ts is the sampling period.

From the above equation, the maximum signal amplitude will be.

Signal power is given as $P = \frac{V^2}{R}$

Here V is the r.m.s. value of the signal. Here $V = \frac{A_m}{\sqrt{2}}$

Peak amplitude becomes, signal power is obtained by taking $R = 1$. Hence above equation signal power is obtained in decibels. With substitution for $A_m$, becomes:

$P = \frac{\delta^2}{8\pi^2 f^2 T_s^2}$

This is an expression for signal power in delta modulation.

(i) To obtain noise power

Uniform distribution of quantization error in delta
We know that the maximum quantization error in delta modulation is equal to step size $\delta$. Let the quantization error be uniformly distributed over an interval $[-\delta, \delta]$ From this figure, the PDF of quantization error is expressed as,

$f(\varepsilon) = \begin{cases} \frac{1}{2\delta} &amp; \text{ for } -\delta &lt; \varepsilon &lt; \delta \ 0 &amp; \text{ otherwise} \end{cases}$

The noise power is given as,

$V<em>{noise}^2 = R = E[\varepsilon^2] = \int</em>{-\infty}^{\infty} \varepsilon^2 f(\varepsilon) d\varepsilon = … = \frac{\delta^2}{3}$

Hence, noise Power will be.

Noise power can be obtained with $R = 1$.

Hence, noise power is: Noise power = $\delta^2/3$

This noise power is uniformly distributed over to frange. This is illustrated in Fig. At the output of the delta modulator receiver, there is a lowpass reconstruction filter whose cut-off frequency is 'W. This cut-off frequency is equal to the highest signal frequency. The reconstruction filter passes part of the noise power at the output as Fig.

From the geometry of Fig. output noise power will be

Output noise $= \frac{W}{f_s} \times noise power$

$f= \frac{1}{T_s}$

Hence the above equation becomes

Output noise power = $\frac{W T_s \delta^2}{3}$

Therefore, signal to noise power ratio at the output of the delta modulation receiver is given as,

$\frac{signal \space power}{noise \space poter}$

$\frac{\delta^2}{8\pi^2 f^2 T_s^2} = \frac{8 \pi^2 f^2}{3W}$

This is an expression for the signal-to-noise power ratio in delta modulation.

Advantages

• Transmission bandwidth is quite small

• Transmitter and receiver implementation is very simple

Disadvantages

• Slope overload distortion (startup error)

• Granular noise (hunting)

Adaptive DM (ADM)

Adaptive delta modulation is a delta modulation system where the step size of the DAC is automatically varied, depending on the amplitude characteristics of the analog input signal.

When the output of the transmitter is a string of consecutive 1s or 0s, this indicates that the slope of the DAC output is less than the slope of the analog signal in either the positive or the negative direction. Essentially, the DAC has lost track of exactly where the analog samples are, and the possibility of slope overload occurring is high.

With ADM, after a predetermined number of consecutive 1s or 0s, the step size is automatically increased. After the next sample, if the DAC output amplitude is still below the sample amplitude, the next step is increased even further until eventually the DAC catches up with the analog signal.

When an alternative sequence of 1s and 0s is occurring, this indicates that the possibility of granular noise occurring is high. Consequently, the DAC will automatically revert to its minimum step size and, thus, reduce the magnitude of the noise error.

A common algorithm for an adaptive delta modulator is when three consecutive 1s or 0s occur, the step size of the DAC is increased by a factor of 1.5.

Various other algorithms may be used for adaptive delta modulators,