Classification of Signals - 1.1

Continuous-Time Signals (CT):

• Definition: A signal is continuous-time if its independent variable, which is usually time (t), can take on any real value within a given interval. This means the signal exists and has a defined value at every single instant within its duration.
• Representation: Denoted as x(t). The "t" in parentheses signifies that the signal is a function of a continuous variable.
• Nature: Often arise from natural physical processes that evolve smoothly over time. Think of it as a smooth, unbroken curve on a graph.

Examples:
- Analog Audio: The sound waves hitting your ear or a microphone produce a voltage signal that changes continuously with time.
- Temperature Readings: The temperature in a room, if measured perfectly, would change continuously.
- Voltage Across a Capacitor: As a capacitor charges or discharges, its voltage changes smoothly over time.
- Speech Signals: The variation in air pressure representing spoken words is a continuous-time signal.

• Graphical Representation: A solid line or curve without any gaps or breaks, extending over the time axis.

Discrete-Time Signals (DT):

• Definition: A signal is discrete-time if its independent variable, typically represented by an integer 'n', can only take on specific, discrete integer values. This means the signal is only defined at particular, separated points in time, not continuously.
• Representation: Denoted as x[n]. The square brackets around 'n' specifically indicate a discrete-time signal.
• Nature: Often result from sampling continuous-time signals (converting analog to digital) or from processes that are inherently discrete (e.g., daily measurements, counts).

Examples:
- Sampled Audio: A compact disc (CD) stores audio by taking samples of the continuous audio signal at a rate of 44,100 samples per second. The signal only exists at these discrete time points.
- Daily Stock Prices: A stock's price is recorded once a day, resulting in a sequence of discrete values.
- Digital Images: An image is a 2D discrete-time signal where pixel values are defined at discrete spatial coordinates.
- Population Counts: The population of a city measured annually.

• Graphical Representation: Individual vertical lines (stems) at integer locations on the horizontal axis, representing the value of the signal at those specific discrete points. The space between these points is empty.

Key Distinction:

The primary difference lies in the nature of the independent variable: continuous (t) for CT signals and discrete (n, integers) for DT signals.

Analog Signals:

• Definition: An analog signal is one whose amplitude (the dependent variable) can take on any value within a continuous range. There are infinitely many possible values the amplitude can assume.
• Nature: Often naturally occurring and perfectly mimics the physical quantity it represents.
• Relationship to CT/DT: While most analog signals are also continuous-time (e.g., microphone output voltage), it's conceptually possible to have a discrete-time analog signal if its samples could still take on any real value (e.g., floating-point samples of a continuous signal before quantization). However, in practical engineering, "analog" usually implies continuous amplitude and continuous time.

Examples:
- Sound Waves: The actual pressure variations in the air are analog.
- Light Intensity: The brightness of light is analog.
- Output of a Thermistor: A sensor whose resistance changes continuously with temperature, producing an analog voltage.

Digital Signals:

• Definition: A digital signal is one whose amplitude is quantized, meaning it can only take on a finite set of discrete values. This process is called quantization.
• Nature: Always discrete in amplitude. They are typically also discrete-time signals, as a continuous-time signal with discrete amplitudes is less common in practical applications.
• Relationship to CT/DT: Digital signals are almost always discrete-time signals. The process of converting an analog (continuous-time, continuous-amplitude) signal to a digital (discrete-time, discrete-amplitude) signal involves two steps: sampling (to make it discrete-time) and quantization (to make its amplitude discrete).

Examples:
- Binary Data (0s and 1s): The most common form of digital signal, used in computers.
- Sampled and Quantized Audio/Video: The data stored on a CD or in an MP3 file, where each sample's amplitude is rounded to the nearest allowed digital value.
- Digital Logic Levels: The voltage levels in a computer circuit representing "high" (e.g., 5V) or "low" (e.g., 0V).

Key Distinction:

The primary difference lies in the nature of the dependent variable (amplitude): continuous values for analog, discrete values for digital.

Periodic Signals:

• Definition: A signal is periodic if its shape or pattern repeats exactly after a fixed interval of time or a fixed number of samples. This repetition extends infinitely in both the positive and negative directions of the independent variable.
• Continuous-Time Periodic Signal: A signal x(t) is periodic if there exists a positive, non-zero constant T0 (called the fundamental period) such that x(t) = x(t + T0) for all values of t. The fundamental period T0 is the smallest such positive constant.
• Related Concepts: The fundamental frequency f0 = 1/T0 (measured in Hertz) and the fundamental angular frequency ω0 = 2π/T0 (measured in radians per second).

Examples:
- A perfect sine wave (sin(ωt)), a square wave that repeats, the voltage from an AC power outlet.

• Discrete-Time Periodic Signal: A signal x[n] is periodic if there exists a positive, non-zero integer N0 (called the fundamental period) such that x[n] = x[n + N0] for all integer values of n. The fundamental period N0 is the smallest such positive integer.
• Key Condition for DT Sinusoids: For a discrete-time sinusoidal signal (e.g., A cos(Ω0 n + φ)), for it to be periodic, the ratio Ω0 / (2π) must be a rational number. That is, Ω0 / (2π) = k/N, where k and N are integers. The fundamental period N0 is then found by reducing k/N to its lowest terms. Unlike CT sinusoids, not all DT sinusoids are periodic.

Examples: The sequence [1, 0, -1, 0, 1, 0, -1, 0, ...], where N0 = 4.

Aperiodic Signals:

• Definition: A signal that does not repeat its pattern over any finite interval of time or number of samples. Their shape is unique and does not recur.
  Examples:
• A single pulse (like a cough sound), a decaying exponential (e.g., the discharge of a capacitor), a transient response in a circuit, a typical segment of speech. Most real-world signals that convey information are aperiodic.

Energy Signal:

• Definition: A signal is an energy signal if its total energy (E) is finite and non-zero (0 < E < infinity). Its average power (P) must be zero. Energy signals typically have finite duration or their amplitude decays to zero as time (or index) approaches positive or negative infinity.
• Total Energy for Continuous-Time Signal x(t): E = integral from -infinity to +infinity of |x(t)|^2 dt.
• Total Energy for Discrete-Time Signal x[n]: E = sum from n=-infinity to n=+infinity of |x[n]|^2.

Examples:
- A single rectangular pulse of finite duration.
- A decaying exponential, such as e^(-at)u(t) for a > 0 (it eventually dies out).
- The impulse function.

Power Signal:

• Definition: A signal is a power signal if its average power (P) is finite and non-zero (0 < P < infinity), while its total energy (E) is infinite. Power signals typically persist indefinitely over time, like periodic signals or random signals.
• Average Power for Continuous-Time Signal x(t): P = limit as T approaches infinity of (1 / (2T)) * integral from -T to +T of |x(t)|^2 dt.
• Average Power for Discrete-Time Signal x[n]: P = limit as N approaches infinity of (1 / (2N + 1)) * sum from n=-N to n=+N of |x[n]|^2.

Examples:
- Any periodic signal, like a sine wave (sin(ωt)) or a square wave. These signals have finite average power but infinite total energy because they continue forever.
- A constant signal (e.g., x(t) = 5).
- White noise (a type of random signal).

Neither Energy nor Power:

Some signals do not fit into either category. For instance, a signal that grows infinitely large over time (e.g., x(t) = e^t * u(t)) would have infinite energy and infinite average power.

Even Signal:

• Definition: A signal x(t) or x[n] is even if it is perfectly symmetric about the time origin. If you fold the signal along the vertical axis, the two halves perfectly overlap.
• Mathematical Condition: x(t) = x(-t) for all t, or x[n] = x[-n] for all n.

Examples:
- The cosine function: cos(ωt) = cos(-ωt).
- A parabolic function: t^2 = (-t)^2.
- The unit impulse function, δ(t) or δ[n].

Odd Signal:

• Definition: A signal x(t) or x[n] is odd if it is anti-symmetric about the time origin. If you fold the signal along the vertical axis and then invert it vertically, the two halves perfectly overlap.
• Mathematical Condition: x(t) = -x(-t) for all t, or x[n] = -x[-n] for all n. Note that for an odd signal, x(0) or x[0] must be 0 if it is defined.

Examples:
- The sine function: sin(ωt) = -sin(-ωt).
- A cubic function: t^3 = -(-t)^3.

Decomposition Property:

Any arbitrary signal (whether CT or DT) can be uniquely decomposed into a sum of an even component and an odd component. This is a very useful property in signal analysis, especially in Fourier series and transforms.
- Even part of x(t): x_e(t) = (1/2) * [x(t) + x(-t)]
- Odd part of x(t): x_o(t) = (1/2) * [x(t) - x(-t)]
- So, x(t) = x_e(t) + x_o(t). The same formulas apply for discrete-time signals x[n].

Deterministic Signals:

• Definition: Signals whose behavior and values are precisely known and can be described by an explicit mathematical function for any given time. There is no uncertainty in their future values.

Examples:
- A perfect sine wave, a step function, an exponential decay. These are the signals we typically analyze using direct mathematical equations.

Random Signals (Stochastic Signals):

• Definition: Signals whose values cannot be precisely predicted and are subject to an element of chance or probability. Their behavior can only be described in terms of statistical averages, probabilities, or ensemble properties.

Examples:
- Thermal noise generated in electronic circuits, speech signals (while deterministic in a very short segment, their overall structure and future values are unpredictable), images (pixel values often exhibit random characteristics). The study of random signals falls under the domain of Probability and Stochastic Processes.