Huiswerkopgaven 5b - Review questions
8 important questions on Huiswerkopgaven 5b - Review questions
Name three types of analog pulse modulation and, for each, state which parameter of the pulse train is varied in proportion to the message signal
a) Pulse Amplitude Modulation (PAM) – the amplitude of each pulse varies with the sample value.
b) Pulse Width Modulation (PWM) – the duration (width) of each pulse
varies with the sample value.
c) Pulse Position Modulation (PPM) – the position of each pulse (relative to a reference time) varies with the sample value
Explain how Time Division Multiplexing (TDM) allows several signals to share a single transmission channel. What property of pulse modulation makes TDM possible
In TDM, each message signal is sampled and assigned a short time slot within a repeating frame. Because pulse modulation uses only a fraction of the channel time for each signal, the gaps between pulses can be used by other signals. Provided the pulse width is short enough, many signals can be interleaved without overlap, all sharing the same physical medium
List the three main processing steps that convert an analog signal into a PCM bit stream, and briefly describe the purpose of each step
1. Sampling – the continuous-time signal is sampled at or above the Nyquist rate to produce a discrete-time sequence of sample values (PAM signal).
2. Quantization – each sample value is approximated by the nearest level from a finite set of L discrete amplitude levels, introducing a controlled quantization
error.
3. Encoding – each quantized level is represented by a binary code word of n = log2 L bits, producing the final PCM bit stream
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In uniform quantization, the difference between the input signal and the quantized output is called the quantization error. Why is this error commonly modeled as noise in system analysis?
Quantization error is modeled as noise because, under typical conditions, the error behaves like an unpredictable disturbance added to the signal. When the input signal varies smoothly relative to the step size, the quantization error appears statistically similar to a random variable with zero mean and bounded amplitude. Treating it as noise simplifies analysis and allows the use of familiar signal-to-noise ratio (SNR) and power-based methods to evaluate system performance
Write down the SNR expression for a uniform L-level PCM quantizer with peak input amplitude mp and signal power S. Using this result, show that every additional bit of resolution increases the SNR by approximately 6 dB.
For a uniform quantizer with L levels and step size ∆ = 2mp/L, the quantization noise power is σ_q^2 = ∆2/12 = m_p^2/(3L^2). The SNR is therefore
The first term increases by 6.02 dB for every additional bit, regardless of the signal. The second term is a constant that depends on the signal statistics; for a full-scale sinusoid (S = m_p^2 /2) it equals 10 log10(3/2) ≈ 1.76 dB.
Explain the motivation for non-uniform (companded) quantization. What type of signal benefits most from this approach, and why
With uniform quantization, the SNR is proportional to signal power S. For signals (such as speech) with a wide dynamic range, this means the SNR is poor during quiet passages. Non-uniform quantization assigns finer (smaller) steps at low amplitudes and coarser steps at high amplitudes, producing a more nearly constant SNR across
the full amplitude range. This is implemented practically via μ-law (North America) or A-law (Europe) companding.
Sigma-Delta (Σ∆) modulation is a refinement of Delta modulation. Describe the key difference in the transmitter architecture and state two advantages this provides
In Σ∆ modulation, the message signal is integrated before being applied to the delta modulator. Equivalently, the integrator can be moved to the feedback path in the receiver. Advantages include:
1. Quantization noise is differentiated at the output, so it does not accumulate (no integration at the receiver).
2. The integrator pre-emphasises low-frequency content (factor 1/jω), making successive samples more correlated and reducing encoding error; slope overload is also less likely because the integrated signal is smoothe
A noiseless channel of bandwidth B uses M signal levels. Write the Nyquist formula for the maximum symbol rate and the resulting maximum bit rate. Explain why using more levels does not always improve practical performance
By Nyquist’s theorem the maximum symbol rate (baud) over a bandwidth B is 2B symbols per second. With M levels, each symbol carries log2 M bits, so the maximum bit rate is
Increasing M raises Rb but also requires the receiver to distinguish more closely spaced signal levels. In the presence of noise (real channels), larger M increases the probability of symbol errors
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