  Here are the answers to the four EQ problems that appeared in Circuit Cellar 322.

Problem 1: Some time ago (Issue #274), we discussed how theoretical channel capacity is a function of the bandwidth and the signal-to-noise ratio of the channel. For example, an SNR of 40 dB limits the channel to 100 different symbols at best, or about 6.64 bits per symbol. It is tempting to use just the integer part of that number, and use only 64 states to encode 6 bits per symbol. But there is a way to use all 100 symbols and maximize the information bandwidth of the channel. Describe the general approach of how you’d encode binary data to be transmitted through an N-state channel.

Answer 1: In the most general case, you would treat a binary (base 2) message as one giant number. In order to transmit that message through a channel that can only carry N different symbols, you would convert that number to base N and transmit the resulting digits one at a time. In practice, you would break a long data stream into fixed-length blocks and then transmit those blocks one at a time, using the above scheme, possibly adding extra error detecting and/or correcting bits to each block.

Problem 2: As a specific example, a 24-dB SNR would limit a channel to no more than 15 symbols. What would be the most efficient way to stream 8-bit bytes through such a channel?

Answer 2: For the specific case of 8-bit bytes through a 15-symbol channel, you might pick a block length of 20 bytes, after noticing that 1541 is 1.6586 × 1048, just a little bit larger than 2160 = 1.4615 × 1048. Each block of 20 bytes would require 41 symbols to be transmitted, achieving an efficiency of 160/41 = 3.90 bits/symbol, which is very close to the theoretical maximum of 3.91 bits/symbol that’s implied by having 15 states.

Problem 3: When we talk about Kirchoff’s Voltage Law (the sum of voltages around a complete loop is zero) and Kirchoff’s Current Law (the sum of currents into a circuit node is zero), we are ignoring some important real-world effects in order to simplify circuit analysis. What are they?

Answer 3: There are three important effects that the “lumped component” model (the model on which KVL and KCL are based) ignores:

• The fact that any circuit node of nonzero size has some capacitance (the ability to store charge) relative to the world at large.
• The fact that any circuit loop of nonzero size has some inductance (the ability to store energy in a magnetic field).
• The fact that fields propagate at a specific velocity (e.g., at the speed of light in a vacuum).

Note that “parasitic” capacitances and inductances can be explicitly added to a lumped component model (where relevant) in order to bring the analysis closer to reality. However, dealing with propagation speed issues (such as transmission line effects) requires a different kind of analysis. Such effects can only be crudely approximated in a lumped-component model.

Problem 4: When doing a high-level design of a sensor-based system, it is often useful to consider exactly what the “observables” are — quantities that can be measured and acted on. For example, many people are interested in measuring the distance between portable devices based on exchanging radio messages, using a protocol such as Bluetooth or WiFi. What exactly are the observables in such a system, and how might they be used to estimate the distance between the transmitter and the receiver?

Answer 4: There are a number of observables associated with a radio network, including:

• The contents of a message
• The time of arrival of a message
• The direction of arrival of a message

The contents of a message can be used to calculate distance if the transmitter reports its own position in a mutually agreed-upon coordinate system, and the receiver also knows its own position. The time of arrival can be used to calculate distance if the time that the message was transmitted is also known. Again, you can get this from the contents of the message if the transmitter and receiver have adequately synchronized clocks. The direction of arrival can be used (assuming the receiver’s antenna is directional enough) to determine the direction of the transmitter relative to the receiver’s orientation. Measurements from multiple transmitters can establish the receiver’s position (and hence its distance) relative to those transmitters. However, this is easily confused by signal reflections in the environment (multipath). The radio signal strength can also be used to estimate distance, but it is a measurement that depends on many things besides distance that need to be accounted for, such as antenna gain and orientation (at both ends), multipath and RF absorption, transmitter power level calibration. This makes it the least useful (and least accurate) way to measure distance.

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