Energy and Power

The watt is defined as one joule per second J/s so power is energy divided by time and energy is the integral of power. This means power is analogous to distance as energy is to velocity.

dB is a unit-less ratio to compare signals. For power, dB = 10*log(Ps/Pr) where Pr is the reference. For voltage, dB = 20*log(Vs/Vr) because it is a root power level instead of regular power level. This is useful since signal strengths vary logarithmically, not linearly. A 3 dB change corresponds to a 50% gain or reduction.

dBm is used to represent power relative to 1mW. Thus 0 dBm = 1mW. The dBm value can be derived from 10*log(Ps). Similarly, dbW is relative to 1 W.

Bias Voltage

Power needed by a device such as a microphone to function. Batteries may be said to provide bias voltage. A fixed, low DC voltage.

In more detail biasing is the setting of DC operating conditions (current and voltage) of an active device in an amplifier. Many electronic devices, such as diodes, transistors and vacuum tubes, whose function is processing time-varying (AC) signals, also require a steady (DC) current or voltage at their terminals to operate correctly. This current or voltage is called bias. The AC signal applied to them is superposed on this DC bias current or voltage. The operating point of a device, also known as bias point, quiescent point, or Q-point, is the DC voltage or current at a specified terminal of an active device (a transistor or vacuum tube) with no input signal applied. A bias circuit is a portion of the device's circuit which supplies this steady current or voltage.

RMS Root Mean Square

The RMS value of a set of values (or a continuous-time waveform) is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous waveform.

If your set of values is [1,2,3] then the RMS is the square root of (1+4+9)/3, 2.16.

The RMS current value can also be defined as the "value of the direct current that dissipates the same power in a resistor." In other words the RMS is equal to the value of DC power that would produce the same average power dissipation in a given resistive load. It's a way to equate an AC level with a DC level. The RMS is not really the AC average, since that is zero, but it's an effective value. This is because we are squaring the negative AC values so the negative aspect is ignored. This simplifies down to .707 (1/sqrt(2)) times the peak AC voltage, or Vrms = Vp/sqrt(2).

Outlet voltages in the US (110,120) are given in Vrms values. AC meters show the RMS effective value.

Coupling and Common Modes

The term coupling refers to the transfer of energy between two media via physical contact. This setting applies to the transducers of any data acquisition system. AC coupling filters out DC components and normalizes the signal to a mean of zero (so in designs for a signal path that ask for an AC coupling capacitor, this is the same as a DC block). The DC component is also known as the offset in the time domain (not a frequency), and after a Fourier transform to the frequency domain it will be at 0 Hz. Both AC and DC are allowed through with DC coupling. The AC coupling circuit adds a series capacitor and parallel resistor and acts as a high-pass filter.

Common mode and differential mode signals are associated with both op-amps and interference noise in circuits. Common mode voltage gain results from the same signal being given to both the inputs of an op-amp. If both signals flow in the same direction, it creates common mode interference, or noise. Differential mode is the opposite of common mode, in that the direction of the signals are different (meaning one line is P and the other N). Differential interference is dominant at lower frequencies, and as the frequency increases, common mode noise is more active. The following solutions can help reduce differential mode noise:

Scaling

Scaling for waveform data is non-trivial. Keysight's user guide for signal generators says "There is no single scaling value that is optimal for all waveforms. To achieve the maximum dynamic range, select the largest scaling value that does not result in a DAC over-range error." An entire mechanism is devoted to this issue with automatic gain conversion (AGC). One of the difficulties is properly adjusting thresholds for doing detection or qualification on multiple signals, so some kind of dynamic threshold is advised instead of fixed.

Signals Basics

The fundamental frequency is the lowest freq of a periodic waveform. The integer multiples of a fundamental frequency are called harmonics.

What is the frequency response? This is the amplitude of a system output over a range of frequencies, plotted as amplitude dB over frequency. With audio for example, you may want this to be uniform across a wide bandwidth. The frequency response characterizes systems in the frequency domain, just as the impulse response characterizes systems in the time domain. In linear systems, either response completely describes the system and thus have one-to-one correspondence: the frequency response is the Fourier transform of the impulse response.

The Fourier series describes periodic signals by combinations of harmonics and sinusoids. Frequency content in the frequency domain is the spectrum. For practical purposes, all periodic signals can be described by a Fourier series. The series can be represented in three different forms: trig, polar, and exponential.

Per Parseval, you can find the signal power in a periodic signal x(t): P=1/T * integ(0,T) of x(t)^2 which also can be derived similarly in the frequency domain.

This is the Dirac delta function symbol: δ. Describes an impulse function (zero everywhere except at x=0, and has integral value of one) which serves as an abstraction to consider an energy transfer as nearly instantaneous at a macro level.

FFT

The FFT is supposed to be a computationally efficient calculation method for a DFT (Discrete FT). The Xilinx FFT core uses the Cooley-Tukey FFT algorithm, apparently the most common of many options. Under the right conditions the DFT returns a sample version of the DTFT using a systematic computation. The mathematical definition of the DFT is complex number multiplication and addition. The forward transform is called analysis' as in spectral analysis on a discrete-time signal input. The inverse IDFT is called synthesis as it synthesizes the discrete signal x[n] from the spectrum values. Each sum is N terms, so complex multiplies/additions is finite computing of N points.

sum x[n] -> DFT -> X[k]
1/N * sum X[k] -> IDFT -> x[n]

In other words, each point of the resulting transformation is derived from a sum of points (multiplying and adding). The signal periods on each side of the original N-point sequence are called periodic extensions of the signal.

The number of operations for a DFT are N^2 for multiplies and N(N-1) for additions, so virtually the same number of each for large values of N. For example, a 64-point DFT requires 4096 complex multiplies and 4032 complex additions. An FFT is designed to reduce these numbers. There are many FFT algorithms. A radix-2 decimation-in-time (DIT) FFT is the simplest and most common form of the Cooley–Tukey algorithm, which breaks a DFT into smaller DFTs. Recursion is used to reduce runtime to O(N log N) from O(N^2).

Spectral Analysis Terms

Synthesis is to combine periodic signals to obtain another periodic signal. Analysis is the separation of a periodic signal into its periodic components.

spectral analysis = the analysis of a periodic signal by its Fourier series
magnitude spectrum = plots of the magnitude of each harmonic; sketched as positive quantity; two-sided mag shows even symmetry about y-axis
phase spectrum = plots of the phase of each harmonic; two-sided phase is mirrored and flipped about y-axis

Spectral Density

energy spectral density: total signal energy is area of the squared magnitude spectrum, measured as Joules per Hertz (J/Hz); in the frequency domain E = integ(inf) |X(f)|^2 df; computed for finite time, i.e. good for transients or time-limited signals

power spectral density: a continuous function with area equal to total signal power, so a kind of running average power in Watts per Hertz (W/Hz) (not a unique indicator of the time signal)
mathematically done with a train of impulses whose total area (magnitude sums) equals total signal power
applies to a signal existing over all time, but PSD is the spectral energy distribution per unit time, i.e. good for continuous signals

Beamforming

Beamforming or spatial filtering is a signal processing technique used in sensor arrays for directional signal transmission or reception. This is achieved by combining elements in an antenna array in such a way that signals at particular angles experience constructive interference while others experience destructive interference. Beamforming can be used at both the transmitting and receiving ends in order to achieve spatial selectivity. The improvement compared with omnidirectional reception/transmission is known as the directivity of the array.

Numerous applications in radar, sonar, seismology, wireless communications, radio astronomy, acoustics and biomedicine. Adaptive beamforming is used to detect and estimate the signal of interest at the output of a sensor array by means of optimal (e.g. least-squares) spatial filtering and interference rejection.

To change the directionality of the array when transmitting, a beamformer controls the phase and relative amplitude of the signal at each transmitter, in order to create a pattern of constructive and destructive interference in the wavefront. When receiving, information from different sensors is combined in a way where the expected pattern of radiation is preferentially observed.

Various Terminology

Analog signals don't really have latency like digital paths do - they just have the delay through the components.

An antenna, like many two-terminal devices, is a load that can be described by an impedance. It can be purely resistive (like a resistor), meaning current and voltage are in phase. Or it can be reactive (like a capacitor or inductor) with current and voltage being 90 degrees out of phase. Or it can be some combination of both.

Resistive components have in-phase current and voltage. Reactive components have out of phase current and voltage.

Active components require an external source to operate in a circuit, while passive components do not. Active components produce energy or control or modify signals, and passive components store or maintain energy in the form of voltage or current. Resistors, inductors, capacitors = passive components.

Excitation = the application of voltage to an electric device, as an electron-tube circuit, an antenna, or a dynamotor, often for producing a magnetic field in the device.