EEL4514C Communication Systems and Components, Fall 2026, Lecture 28
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Overview
The lecture combines practical sampling and reconstruction distortions into one baseband equalizer, then introduces pulse modulation and pulse-code modulation (PCM). PCM improves robustness to noise but raises bandwidth requirements: examples include 64 kb/s telephone audio and 705.6 kb/s mono CD audio, motivating compression to exploit signal redundancy.
Key takeaways
- For a passband channel of bandwidth B carrying m bits per symbol, the lecture uses Bm bits per second as a practical information-rate estimate; M distinct signaling levels encode log₂(M) bits per symbol.
- Practical sampling and reconstruction distortions can be compensated together by one equalizer, provided their combined frequency response has no zeros within the desired baseband.
- PCM samples, quantizes, and binary-encodes an analog signal; n-bit quantization at 2B samples per second creates a 2Bn bit/s stream that needs roughly n times the original analog bandwidth under binary signaling.
- Telephone PCM at 8 kHz and 8 bits per sample yields 64 kb/s, with an estimated 32 kHz transmission bandwidth compared with a 3.4 kHz voice cutoff.
- Mono CD audio at 44.1 kHz and 16 bits per sample yields 705.6 kb/s; stereo doubles the rate, making compression valuable for reducing the bandwidth required by uncompressed PCM.
- PAM varies pulse amplitude, while PWM and PPM encode information in pulse width or position; constant-amplitude pulses can ease amplifier design, but timing accuracy is especially important for PPM.
Chapters
- For a passband channel of bandwidth B, the lecture estimates a maximum rate of B times m bits per second, where m is bits per symbol.
- A constellation with M signaling levels carries log₂(M) bits per symbol; examples include QPSK with 2 bits and 16-QAM with 4 bits.
- Ideal impulse sampling is physically impractical, so real samplers average the input over a short interval using a rectangular pulse.
- At 8:40, Mingyue Ji introduces the lecture plan: finish practical sampling, then cover pulse modulation and the advantages of digital communication.
- The practical sampler's periodic pulse train is described through Fourier-series coefficients, producing repeated spectral components spaced by the sampling frequency.
- Because reconstruction is also filtering, its response P_F can be combined with the practical-sampling response before equalizer design.
- The desired baseband signal is recovered by considering the zero-index spectral replica, rather than equalizing every higher-frequency component created by sampling.
- The equalizer compensates jointly for the sampling response and reconstruction filter, instead of designing separate equalizers for each effect.
- Within the signal band, the combined sampling and reconstruction response must not be zero, or the equalizer cannot invert it.
- Pulse modulation represents information by changing a rectangular pulse train's parameters, including amplitude, width, or position.
- Pulse-amplitude modulation (PAM) varies pulse heights; pulse-width modulation (PWM) and pulse-position modulation (PPM) keep pulse amplitudes constant.
- PWM and PPM simplify amplitude handling, but PPM in particular requires accurate timing to identify each pulse's information-bearing position.
- Pulse-code modulation (PCM) samples an analog input, quantizes each sample, and encodes the resulting level as binary digits.
- Four bits represent 16 quantization levels, while three bits represent 8 levels; the lecture's signal-processing chain includes an anti-aliasing low-pass filter, sampling, quantization, and a bit encoder.
- A binary waveform can transmit each bit, for example using no pulse for 0 and a rectangular pulse for 1.
- Pulse-based digital signals can interleave transmissions from sources such as G1 and G2 in time, making them distinguishable at the receiver when timing is synchronized.
- At the Nyquist sampling rate of 2B samples per second, n-bit PCM produces a bit rate of 2Bn bits per second.
- With binary signaling, that rate requires approximately nB bandwidth, about n times the original analog signal's bandwidth under the lecture's rate-to-bandwidth model.
- The telephone example low-pass filters voice at a 3.4 kHz cutoff and samples at 8,000 samples per second, leaving margin above the 6.8 kHz Nyquist minimum.
- Using 8 bits per sample produces 8,000 × 8 = 64,000 bits per second, or 64 kb/s.
- The lecture estimates about 32 kHz of transmission bandwidth for this binary PCM stream, roughly ten times the 3.4 kHz voice bandwidth.
- The CD example limits music to about 20 kHz and uses a 44.1 kHz sampling rate, exceeding the 40 kHz Nyquist minimum.
- Sixteen-bit quantization yields 44,100 × 16 = 705,600 bits per second, or 705.6 kb/s for mono audio; stereo requires twice that rate.
- The resulting uncompressed bitstream needs hundreds of kilohertz of channel bandwidth under the lecture's binary signaling estimate, far more than the 20 kHz audio band.
- PCM's telephone and CD examples illustrate the tradeoff: digital signals can be more robust to noise, but their uncompressed bitstreams consume substantially more bandwidth.
- Binary decisions can let a receiver recover a transmitted 0 or 1 despite noise, unlike direct recovery of an arbitrarily distorted analog waveform.
- The lecture identifies compression as the way to reduce digital bit rate by exploiting redundancy, such as small differences between successive audio samples.
- Repeated or slowly changing signal values contain redundancy, so a codec can send a compact representation of changes instead of retransmitting every sample in full.
- Mingyue Ji notes that compression can dramatically shrink data, giving a possible reduction to one-hundredth of the original size as an illustration.
- Reducing the bit rate lowers the required transmission bandwidth and helps offset PCM's main disadvantage.
Transcript excerpt
0:04 Okay, let's get started. Yeah, let's see what we cover in our last lecture. In our last lecture, uh we first talk about the maximum information rate in uh band limited channels and then we talk about non ideal or practical sampling, right? So before we talk about the practical reconstruction, so here is the practical sampling. Okay. So basically uh uh the way to see what is the maximum information rate inbound limit channel is via the network sampling rate right. So basically for a bassband ch for a bband signal right and uh we consider if we use a nequis sampling rate so basically what we're going to get is two b symbols per second right and if we consider is a rf channel or is a pathband channel. Uh so basically uh the bandwidth is basically
0:55 B instead of 2B here. Okay. And uh so what we're going to do is that we're going to use M different uh M different capital M different channels for the modulation right and then we can transmit for every sample. So we basically can transmit uh lowerase M bits per symbols. Okay. So here is an example that let's say we have eight levels of signaling right and then so basically for each uh signal we can repres use three bits to represent it. So those are basically uh eight levels. Okay, it's kind of like a quantization for example, right? We just transmit those levels, right? And um and here uh are some practical uh signaling that we're using right for digital communication normally we use those kind of signaling right qpsk 16 quam 6 uh 64
1:45 quam and 256 quam and so on right so we can write it so now I think state would be 1024 quam right for for so this so q
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Related notes
Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Mingyue Ji.