Аналоговая обработка сигналов: основы и методы.

Analog signal processing is a type of signal processing conducted on continuous analog signals by some analog means (as opposed to the discrete digital signal processing where the signal processing is carried out by a digital process). "Analog" indicates something that is mathematically represented as a set of continuous values. This differs from "digital" which uses a series of discrete quantities to represent signal. Analog values are typically represented as a voltage, electric current, or electric charge around components in the electronic devices. An error or noise affecting such physical quantities will result in a corresponding error in the signals represented by such physical quantities. Examples of analog signal processing include crossover filters in loudspeakers, "bass", "treble" and "volume" controls on stereos, and "tint" controls on TVs. Common analog processing elements include capacitors, resistors and inductors (as the passive elements) and transistors or opamps (as the active elements).

A system's behavior can be mathematically modeled and is represented in the time domain as h(t) and in the frequency domain as H(s), where s is a complex number in the form of s=a+ib, or s=a+jb in electrical engineering terms (electrical engineers use "j" instead of "i" because current is represented by the variable i). Input signals are usually called x(t) or X(s) and output signals are usually called y(t) or Y(s).

Convolution is the basic concept in signal processing that states an input signal can be combined with the system's function to find the output signal. It is the integral of the product of two waveforms after one has reversed and shifted; the symbol for convolution is *. That is the convolution integral and is used to find the convolution of a signal and a system; typically a = ∞ and b = +∞. Consider two waveforms f and g. By calculating the convolution, we determine how much a reversed function g must be shifted along the x axis to become identical to function f. The convolution function essentially reverses and slides function g along the axis, and calculates the integral of their (f and the reversed and shifted g) product for each possible amount of sliding. When the functions match, the value of (f*g) is maximized. This occurs because when positive areas (peaks) or negative areas (troughs) are multiplied, they contribute to the integral.

Bode plots are plots of magnitude vs. frequency and phase vs. frequency for a system. The magnitude axis is in [Decibel] (dB). The phase axis is in either degrees or radians. The frequency axes are in a [logarithmic scale]. These are useful because for sinusoidal inputs, the output is the input multiplied by the value of the magnitude plot at the frequency and shifted by the value of the phase plot at the frequency.

This is the domain that most people are familiar with. A plot in the time domain shows the amplitude of the signal with respect to time.

A plot in the frequency domain shows either the phase shift or magnitude of a signal at each frequency that it exists at. These can be found by taking the Fourier transform of a time signal and are plotted similarly to a bode plot.

While any signal can be used in analog signal processing, there are many types of signals that are used very frequently.

Sinusoids are the building block of analog signal processing. All real world signals can be represented as an infinite sum of sinusoidal functions via a Fourier series. A sinusoidal function can be represented in terms of an exponential by the application of Euler's Formula.

An impulse (Dirac delta function) is defined as a signal that has an infinite magnitude and an infinitesimally narrow width with an area under it of one, centered at zero. An impulse can be represented as an infinite sum of sinusoids that includes all possible frequencies. It is not, in reality, possible to generate such a signal, but it can be sufficiently approximated with a large amplitude, narrow pulse, to produce the theoretical impulse response in a network to a high degree of accuracy. The symbol for an impulse is δ(t). If an impulse is used as an input to a system, the output is known as the impulse response. The impulse response defines the system because all possible frequencies are represented in the input

Linearity means that if you have two inputs and two corresponding outputs, if you take a linear combination of those two inputs you will get a linear combination of the outputs. An example of a linear system is a first order low pass or high pass filter. Linear systems are made out of analog devices that demonstrate linear properties. These devices don't have to be entirely linear, but must have a region of operation that is linear. An operational amplifier is a non linear device, but has a region of operation that is linear, so it can be modeled as linear within that region of operation. Time invariance means it doesn't matter when you start a system, the same output will result. For example, if you have a system and put an input into it today, you would get the same output if you started the system tomorrow instead. There aren't any real systems that are LTI, but many systems can be modeled as LTI for simplicity in determining what their output will be. All systems have some dependence on things like temperature, signal level or other factors that cause them to be non linear or non time invariant, but most are stable enough to model as LTI. Linearity and time invariance are important because they are the only types of systems that can be easily solved using conventional analog signal processing methods. Once a system becomes non linear or non time invariant, it becomes a non linear differential equations problem, and there are very few of those that can actually be solved. (Haykin & Van Veen 2003)