Signals: Are functions of one or more independent variables and typically contain information about the behavior or nature of some phenomenon. Systems usually respond to particular signals by producing other signals.

Electrical brain activity => Power Lab => voltage variations in time Signal System Signal

Information in a signal is contained in a pattern of variations of some form. E.g. the human vocal mechanism produces speech by creating fluctuations in acoustic pressure.

Deterministic Signals: A signal is deterministic if it is exactly predictable for the time span of interest. Deterministic signals can be described by mathematical models, e.g., a sinusoidal signal is described by: V(t) = A * sin(ω*t), where V(t) is the signal over time. ‘A’ is the amplitude and ω= 2πf (f= frequency of the signal).

Stochastic or Random Signals: A signal whose value has some element of chance associated with it, therefore it cannot be predicted exactly. Consequently, statistical properties and probabilities must be used to describe stochastic signals.

Usually, biological signals often have both deterministic and stochastic components.

Desired Signal: A signal that it is not corrupted by noise.

Signal Amplitude Statistics: A number of statistics may be used as a measure of the location or “center" of a random signal.

The mean is the average amplitude of the signal over time.

The median is the value at which half of the observations in the sample have values smaller than the median and half have values larger than the median. The median is often used as a measure of the “center" of a signal because it is less sensitive to outliers.

The mode is the most frequently occurring value of the signal.

Maximal and minimal amplitude are the maximal and minimal maximal values of the signal during a given time interval.

Range: The range or peak-to-peak amplitude is the difference between the minimum and maximum values of a signal.

Noise: Any unwanted signal that modifies the desired signal. It could have multiple sources.

Signal to Noise Ratio (SNR): It is a measurement of the amplitude of variance of the signal relative to the variance of the noise. The higher the SNR, the better you can distinguish your signal from the noise.

Noise sources: Any discussion of filtering for noise reduction would be incomplete without some discussion of noise. Let’s start by defining some common types of noise.

Thermal noise – the random motion of atoms generates this random, uniformly distributed noise. Thermal Noise is present everywhere and has a nearly constant Power Spectral Density (PSD).

Interference – imposition of an unwanted signal from an external source on the signal of interest.

Aliasing – an artifact of the acquisition process, specifically sampling (see Nyquist rate).

Sampling noise – Another artifact of the acquisition process, Sampling Noise occurs when you digitize a continuous signal with an A/D converter that has a finite number of steps. It is interesting to note that you can dither (add white noise) your signal to reduce the overall sampling noise.

Narrowband/broadband – two general categories of noise. Narrowband noise confines itself to a relatively small portion of the overall signal bandwidth as defined by Nyquist. Broadband noise occupies a significant portion of the Nyquist bandwidth. For example, 60-Hz hum is narrowband because it typically limits itself to a 60 Hz component. Thermal noise is definitely broadband because its PSD is constant, meaning that it distributes its energy over nearly the entire spectrum.

Waveform: The representation of a signal as a plot of amplitude versus time

Continuous time signals: The independent variable is continuous, the signals are defined for a continuum of values of the independent variable X(t).

Discrete time signals: Only defined at discrete times, the independent variable takes on only a discrete set of values X(n). A discrete time signal may represent a phenomenon for which the independent variable is inherently discrete (e.g., amount of calories per day on a diet). On the other hands, a discrete signal may represent successive samples of an underlying phenomenon for which the independent variable is continuous (e.g., a visual image capture by a digital camera is made of individual pixels that can assume different colors).

An analog signal is a continuous time signal.

A digital signal is a discrete time signal.

Analog-digital converters (ADC): It is a system that inputs an analog electrical signal such as voltage or current and outputs a binary number (0 or 1). The computer's ADC allows an electrical signal to be sampled and converted into a digital signal, which is then sent within the computer for further processing.

Signal sampling: The process of obtaining a sequence of instantaneous values of a particular signal characteristic, usually at regular time intervals.

Sampling frequency: The sampling frequency is the frequency at which the ADC samples the analogue signal (usually in number of samples per second, (Hz)).

Sampling Period: The reciprocal of the sampling frequency, i.e., the interval between corresponding points on two successive sampling pulses of the sampling signal.

Sampling Range: The range between the minimal and maximal values at which you will sample the signal (e.g., if you sample between -10 V and +10 V the sampling range is 20V)

Offset: A fluctuation in the baseline value of the signal.

Gain and amplification: It is the factor by which you multiply your signal. If a gain is 1, the signal remains unchanged, if the gain is higher than 1, the signal is amplified, if the gain is lower than 1, the signal is reduced.

Amplitude saturation: It occurs when the intensity of a signal exceeds the values within the sampling range. For example if we acquire a signal which intensity is +20V and we are sampling between -5V and +5V. It produces a distortion of the signal, i.e., over the interval in which the signal reaches the +20V, the output of our ADC will be +5V.

Spectral analysis: Is the process of decomposing a signal in different frequency components and plot the intensity of each component as a function of its frequency.

Fourier analysis: It is a mathematical technique that allows us to perform a spectral analysis on the recorded signal.

Nyquist interval: The maximum time interval between equally spaced samples of a signal that will enable the signal waveform to be completely determined. The Nyquist interval is equal to the reciprocal of twice the highest frequency component of the sampled signal.

Nyquist Sampling Rate: Is the value of the sampling frequency equal to twice the maximal frequency of the signal we are acquiring.

Filters: are the devices that alter the frequency composition of the signal. The bandwidth or range of the signal frequencies is determined by filters.

Ideal Frequency-selective filter: Is a filter that exactly passes signals at one frequency and completely rejects the rest.

There are three types of filters:
1. Low frequency or in old terminology high pass. Filters low frequencies
2. High frequency or in old terminology low pass. Filters high frequencies.
3. Notch filter. Filters one frequency, usually 60 Hz from the power lines.

Real filters or hardware filters alter the frequency composition of the signal. It means after filtering the signal, we cannot recover the frequencies that have been filtered.

Digital filters change the frequency of the signal by performing calculations on the data. It means you can record all the frequency components of your signal and by digitally filtering it, eliminate the unwanted frequencies. You can still recover the filtered frequencies if you keep a record of the original signal.