**Continuous Time Signal** are mathematical representations of signals that **vary continuously over time.** They are **defined and observed at every instant of time within a specified interval.** In other words, a continuous-time signal is **defined for an infinite number of points in time**. Continuous & Discrete Time Signal:

A continuous-time signal can take on various forms and can represent a wide range of physical quantities or phenomena. Some examples of continuous-time signals include:

**Analog audio signals:**Continuous-time signals are used to represent sounds in the form of electrical voltage variations. For example, the waveform produced by a microphone capturing someone’s voice or a musical instrument is a continuous-time signal.**Voltage waveforms:**In electrical engineering, voltage signals that vary continuously over time are represented as continuous-time signals. These can include periodic waveforms like sine waves, square waves, and triangular waves, as well as non-periodic signals.**Temperature measurements:**Temperature variations over time, such as the temperature of a room or the temperature of a chemical process, can be represented as continuous-time signals.**Biomedical signals:**Signals related to physiological processes, such as electrocardiogram (ECG) and electroencephalogram (EEG), are continuous-time signals that represent the electrical activity of the heart and brain, respectively.**Sensor measurements:**Signals obtained from various sensors, such as pressure sensors, humidity sensors, and motion sensors, can be represented as continuous-time signals.

Mathematically, continuous-time signals are typically represented by functions of time, denoted as x(t), where t represents the continuous time variable. These signals can be further analyzed, processed, and manipulated using various techniques and tools in continuous-time signal processing, such as Fourier analysis, filtering, modulation, and system modeling.

It’s important to note that continuous-time signals are often converted into discrete-time signals for digital processing and analysis using techniques such as sampling, where the continuous-time signal is sampled at discrete intervals to obtain a discrete representation of the original signal.

Sine Wave Plot

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**Discrete time Signal**

Discrete time signal is defined at specific points or intervals in time. These signals are typically represented as sequences of values. Examples of discrete-time signals include digital audio signals, sampled temperature measurements, and binary sequences.

Discrete-time systems are mathematical models that operate on discrete-time signals. They can be represented by difference equations or difference equations in the z-domain. These systems take discrete-time signals as input and produce discrete-time signals as output. Examples of discrete-time systems include digital filters, digital controllers, and digital communication systems.

**Representation of Continuous-Time Signals and Systems**

Continuous-time signals can be represented using mathematical functions, such as:

- Time-domain representation: x(t), where t represents the continuous time variable.
- Frequency-domain representation: X(f), where f represents the continuous frequency variable.

Continuous-time systems can be represented by differential equations or transfer functions, such as:

- Differential equation representation: dx(t)/dt + a1x(t) + a0x(t) = b0u(t), where dx(t)/dt represents the derivative of x(t) with respect to time, and u(t) represents the input.
- Transfer function representation: H(s), where s represents the complex frequency variable.

Discrete-time signals can be represented using sequences of values, such as:

- Time-domain representation: x[n], where n represents the discrete time index.
- Frequency-domain representation: X[k], where k represents the discrete frequency index.

Discrete-time systems can be represented by different equations or transfer functions in the z-domain, such as:

- Difference equation representation: y[n] = a1y[n-1] + a0y[n-2] + b0u[n], where y[n] represents the output, u[n] represents the input, and a0, a1, and b0 are coefficients.
- Transfer function representation: H(z), where z represents the complex variable in the z-domain.

It’s important to note that these representations are just a few examples, and there are various other mathematical representations and transformations used to analyze and design continuous-time and discrete-time signals and systems.

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