Signals and systems

Signals and Systems

• Signals are functions of time or other independent variables (e.g., spatial coordinates) carrying information.
• Systems are operations or processes that transform signals.
• Signals can be categorized as continuous-time or discrete-time, based on whether they are defined for all values of the independent variable (e.g., all real numbers for time) or only at specific points (e.g., integers for time).
• Another type of signal is a digital signal (discrete-time and discrete-amplitude) that uses sampled and quantized values.
• Approximations: Trigonometric Fourier series, and Exponential Fourier series are methods used to approximate a function given on some interval.
• Signal operations can enhance, extract, or correct signals.
• Important signal characteristics: independent variables, more than one variable, random, bandwidth.
• Band width: Every signal must be specified with some bandwidth.
• Function of a signal is a mathematical representation of the signal. A graph of a signal is a visual representation of the data and/or signal as a function of time and amplitude.
• Transformation: CTFT (Continuos Time Fourier Transform), LT (Laplace Transform). DTFT (Discrete Time Fourier Transform), ZT (Z-transform), etc.
• Examples of signals: speech, ECG, image.
• Digital signals are discrete-time and discrete-amplitude signals, represented by a sequence of quantized samples.
• Continuous-time signals have a continuous variation of amplitude over time.

Sampling

• Sampling is the process of converting a continuous-time signal into a discrete-time signal by taking samples of the continuous-time signal at regular intervals.
• The sampling rate or frequency (fs) is the number of samples taken per unit of time, typically measured in samples per second (Hz).
• The Nyquist-Shannon sampling theorem states that a continuous-time signal can be perfectly reconstructed from its sampled version if the sampling rate is at least twice the maximum frequency component in the signal.

Discrete-time Signals

• A discrete-time signal is a signal that is defined only at discrete time instances.
• Discrete-time signals are commonly represented as a sequence of values, denoted as x[n], where n represents the discrete-time index.

Digital Signals

• A digital signal is a discrete-time signal with discrete amplitude values.
• These values are often represented by binary digits (bits), leading to the use of binary codes.
• The amplitude values of digital signals can be represented using binary representation

Signal Characteristics

• Multivariable signals: Signals that are functions of more than one independent variable. (e.g., image)
• More than one independent variable: This describes signals having more than one independent variable in their definition, such as an image or a video signal.
• Randomness: Signals that exhibit random or stochastic behavior, often described statistically with properties like mean, variance, and autocorrelation.

Continuous Time Signal

• Continuous variation of amplitude across time.

Discrete Time Signal

• Variation of amplitude at discrete points in time, often using sampled values.

Digital Signal

• Discrete time signal variations, with a discrete amplitude. often used as binary digits (bits) to represent values.

.