WEBVTT NOTE Fourier Analysis, RMS, and Analogue-to-Digital Conversion 1 00:00:00.000 --> 00:00:03.040 Two views of the same signal. 2 00:00:03.040 --> 00:00:09.240 Question. Which of the following describes a signal in the time domain? 3 00:00:10.140 --> 00:00:14.220 Answer. Amplitude at different times. 4 00:00:14.820 --> 00:00:24.820 The oscilloscope view. The frequency domain, the spectrum analyser view, shows amplitude at different frequencies. 5 00:00:24.820 --> 00:00:33.088 Question. What technique shows that a square wave is made up of a sine wave and its odd harmonics? 6 00:00:33.988 --> 00:00:37.468 Answer. Fourier analysis. 7 00:00:38.068 --> 00:00:46.182 Fourier's result is that any periodic waveform is a sum of sine waves at harmonics of its fundamental. 8 00:00:46.182 --> 00:01:10.924 A square wave is the fundamental plus its odd harmonics in declining amplitude, which is not a mathematical curiosity but the explanation for two practical facts: A switching amplifier radiates harmonics, because its output really is a sum of them, which is why the circuits lesson insisted on an output filter. 9 00:01:10.924 --> 00:01:27.948 Sharp CW keying produces clicks, because a fast rise time is a sharp-edged waveform, and sharp edges are made of high harmonics. The FFT from the SDR lesson is the computational version of this transform. True RMS. 10 00:01:27.948 --> 00:01:35.846 Question. What is the benefit of making voltage measurements with a true-RMS calculating meter? 11 00:01:36.746 --> 00:01:44.386 Answer. RMS is measured for both sinusoidal and non-sinusoidal signals. 12 00:01:44.986 --> 00:02:00.456 An ordinary AC meter rectifies, averages, and multiplies by 1.11, a constant that is correct only for a sine wave. On anything else it reads wrong, and it does not warn you. 13 00:02:00.456 --> 00:02:07.609 A true-RMS meter squares, averages, and takes the root, which is the definition. 14 00:02:07.609 --> 00:02:26.388 It reads correctly on a distorted waveform, a switching supply's output, or a modulated signal, exactly the cases where you most want a reliable number. This is the meter worth paying for. PEP-to-average ratio. 15 00:02:26.388 --> 00:02:36.182 Question. What is the approximate ratio of PEP-to-average power in an unprocessed single-sideband phone signal? 16 00:02:37.082 --> 00:02:41.322 Answer. 2.5 to 1. 17 00:02:41.922 --> 00:02:51.281 Question. What determines the PEP-to-average power ratio of an unprocessed single-sideband phone signal? 18 00:02:52.181 --> 00:02:55.122 Answer. Speech characteristics. 19 00:02:55.722 --> 00:03:12.976 Speech is peaky: brief loud syllables between quiet gaps. So an SSB signal at 100 W PEP averages about 40 W, and the ratio depends on the speaker, voice, accent, microphone technique. 20 00:03:12.976 --> 00:03:34.943 Two consequences already met elsewhere: a speech processor raises the average by compressing this ratio, and the duty cycle that matters for RF exposure and for transmitter heating is far lower on SSB than on a constant-carrier mode. 21 00:03:34.943 --> 00:03:53.234 ADC architectures. Three ways to turn a voltage into a number, each with a purpose. Type, Successive approximation; How, binary search, one bit per step; Suits, general purpose, good resolution. 22 00:03:53.234 --> 00:04:12.562 Type, Flash (direct); How, comparators for every level at once; Suits, very high speed, digitizing high frequencies. Type, Sigma-delta; How, oversample and filter; Suits, audio, very high resolution, slow. 23 00:04:12.562 --> 00:04:21.810 Question. Why are direct or flash conversion analog-to-digital converters used for a software defined radio? 24 00:04:22.710 --> 00:04:28.712 Answer. Very high speed allows digitizing high frequencies. 25 00:04:29.312 --> 00:04:50.272 A flash converter decides in one step, because it compares against every level simultaneously. The cost is 2ⁿ−1 comparators, so an 8-bit flash needs 255 of them and a 16-bit one is impractical. Speed bought with silicon. 26 00:04:50.272 --> 00:04:59.402 Question. How many different input levels can be encoded by an analog-to-digital converter with 8-bit resolution? 27 00:05:00.302 --> 00:05:04.142 Answer. 256. 28 00:05:04.742 --> 00:05:19.662 2 to the power of 8 = 256. Same powers-of-two arithmetic as the flip-flop divider chain. Dither, filtering, and quality. 29 00:05:19.662 --> 00:05:26.382 Question. What is "dither" with respect to analog-to-digital converters? 30 00:05:27.282 --> 00:05:35.042 Answer. A small amount of noise added to the input signal to reduce quantization noise. 31 00:05:35.642 --> 00:05:49.463 Counterintuitive and real. A signal sitting between two quantisation levels always rounds the same way, producing a fixed error that appears as distortion. 32 00:05:49.463 --> 00:06:03.642 Adding a little noise makes the rounding vary, so averaging over time recovers information finer than one step. Noise deliberately added to improve resolution. 33 00:06:03.642 --> 00:06:12.322 Question. What is the purpose of a low-pass filter used at the output of a digital-to-analog converter? 34 00:06:13.222 --> 00:06:19.302 Answer. Remove spurious sampling artifacts from the output signal. 35 00:06:19.902 --> 00:06:33.902 A DAC's output is a staircase, full of energy at the sample rate and its harmonics. The reconstruction filter smooths it, the counterpart of the anti-alias filter on the way in. 36 00:06:33.902 --> 00:06:41.552 Question. Which of the following is a measure of the quality of an analog-to-digital converter? 37 00:06:42.452 --> 00:06:45.792 Answer. Total harmonic distortion. 38 00:06:46.392 --> 00:07:03.097 THD captures how faithfully the converter reproduces a signal, which bit depth alone does not: a converter can have 16 bits of resolution and 12 bits' worth of honest performance. Check yourself. 39 00:07:03.097 --> 00:07:18.432 Why does a square wave contain odd harmonics, and what does that predict about a Class D amplifier? Your averaging AC meter reads 10 V on a switching supply's output. Trustworthy? 40 00:07:18.432 --> 00:07:22.030 How many levels does a 12-bit ADC resolve? 41 00:07:28.030 --> 00:07:47.103 Fourier analysis shows a square wave is its fundamental plus odd harmonics. A Class D amplifier's output is a switching waveform, so it radiates those harmonics, hence the mandatory output filter. No. 42 00:07:47.103 --> 00:07:58.165 An averaging meter's 1.11 factor is correct only for a sine wave. Use a true-RMS meter. 2 to the power of 12 = 4096.