Signals and systems By Dr. Amin Danial Asham

References A. V. OPPENHEIM, A. S. WILLSKY and S. H. NAWAB , Signals & Systems, PRENTICE HALL, 1996.

Basic Definitions What is a signal? A signal is a function of one or more independent variables describes a certain phenomenon. A detectable physical quantity or impulse (as a voltage, current, or magnetic field strength) by which messages or information can be transmitted. [Merriam-Webster]

Basic Definitions Examples of signals[http://parlos.tamu.edu/MEEN651/E1.pdf] Current from a current sensor (continuous-time). The variation in pressure in the cylinder of an IC engine during operation (continuous-time). The variation in the number of shoes sold in a certain shop during the month (discrete-time). . The variation in temperature at noon on successive days at a holiday resort (discrete- time).

Basic Definitions What is a System? A regularly interacting or independent group of items forming a unified whole. [Merriam- Webster] Examples of systems: Electric Circuits Electronic circuits Cars Airplanes Rockets Robots

Basic Definitions Systems respond to inputs by producing some kind of outputs in a way that reflects a certain behavior. Inputs and outputs of a system are signals.

Examples Voltages and currents as a function of time in an electrical circuit are examples of signals, and a circuit is itself an example of a system, which in this case responds to applied voltages and currents. An automobile driver depresses the accelerator pedal, the automobile responds by increasing the speed of the vehicle. In this case, the system is the automobile, the pressure on the accelerator pedal the input to the system, and the automobile speed the response A camera is a system that receives light from different sources and reflected from objects and produces a photograph

CONTINUOUS-TIME AND DISCRETE-TIME SIGNALS Continuous-Time Signals: A continuous-time signal is a varying quantity (a signal) whose domain (time) is a continuum. That is, the function's domain is an uncountable set. For continuous-time signals the independent variable is enclosed in parentheses ( · )

CONTINUOUS-TIME AND DISCRETE-TIME SIGNALS Examples of Continuous-Time Signals: Current from a current sensor (continuous- time). The variation in pressure in the cylinder of an IC engine during operation (continuous- time).

CONTINUOUS-TIME AND DISCRETE-TIME SIGNALS A discrete-time signal is a time series consisting of a sequence of quantities. In other words, it is a time series that is a function over a domain of integers. For discrete-time signals the independent variable is enclosed in brackets [ · ].

CONTINUOUS-TIME AND DISCRETE-TIME SIGNALS A discrete-time signal 𝑥[𝑛] may represent a phenomenon for which the independent variable is inherently discrete. A very important class of discrete-time signals arises from the sampling of continuous-time signals. In this case, the discrete-time signal 𝑥[𝑛] represents successive samples of an underlying phenomenon for which the independent variable is continuous.

CONTINUOUS-TIME AND DISCRETE-TIME SIGNALS Examples of Discrete-time Signals: The variation in the number of shoes sold in a certain shop during the month (discrete- time). The variation in temperature at noon on successive days at a holiday resort (discrete-time).

Signal Energy and Power From electrical circuits, the instantaneous power dissipated in a resistor 𝑅 is: 𝑝 𝑡 =𝑣 𝑡 .𝑖 𝑡 = 𝑣(𝑡) 2 𝑅 The total energy over the interval [𝑡1,𝑡2]: 𝐸= 𝑡1 𝑡2 𝑝 𝑡 𝑑𝑡 = 𝑡1 𝑡2 𝑣(𝑡) 2 𝑅 𝑑𝑡 The average power is: 𝑃 𝑎𝑣 = 1 𝑡2−𝑡1 𝑡1 𝑡2 𝑝 𝑡 𝑑𝑡 = 1 𝑡2−𝑡1 𝑡1 𝑡2 𝑣(𝑡) 2 𝑅 𝑑𝑡

Signal Energy and Power Similarly for any continuous –time signal 𝑥(𝑡), the total energy of the signal over [𝑡1,𝑡2] is defined: 𝐸≜ 𝑡1 𝑡2 𝑥(𝑡) 2 𝑑𝑡 (1) where 𝑥 denotes the magnitude of the (possibly complex) number 𝑥. The average power is: 𝑃 𝑎𝑣 ≜ 1 𝑡2−𝑡1 𝑡1 𝑡2 𝑥(𝑡) 2 𝑑𝑡 (2)

Signal Energy and Power In the same way, the total energy of a discrete- time signal 𝑥[𝑛] over the interval 𝑛1≤𝑛≤𝑛2 is defined as: 𝐸≜ 𝑛=𝑛1 𝑛2 𝒙[𝒏] 𝟐 (3) Average power is defined as: 𝑃 𝑎𝑣 ≜ 1 𝑛2−𝑛1+1 𝑛=𝑛1 𝑛2 𝒙[𝒏] 𝟐 (4) No. of point in the interval

Signal Energy and Power It is important to remember that the terms "power" and "energy" are used here independently of whether the quantities in equations (1) to (4) are actually related to physical energy.

Signal Energy and Power Even if a relationship between the signals and physical energy does exist, equations (1) to (4) may have the wrong dimensions and scaling's For example, we see that if 𝒙(𝒕) represents the voltage across a resistor, then 𝑡1 𝑡2 𝑥(𝑡) 2 𝑑𝑡 must be divided by the resistance (measured, for example, in ohms) to obtain units of physical energy.

Signal Energy and Power Total energy and power over infinite interval for a continuous–time signal. Total Energy is: 𝐸 ∞ ≜ lim 𝑇→∞ −𝑇 𝑇 𝑥(𝑡) 2 𝑑𝑡 = −∞ ∞ 𝑥(𝑡) 2 𝑑𝑡 Average power is 𝑃 ∞ ≜ lim 𝑇→∞ 1 2𝑇 −𝑇 𝑇 𝑥(𝑡) 2 𝑑𝑡

Signal Energy and Power Total energy and power over infinite interval for a discrete–time signal. Total Energy is: 𝐸 ∞ ≜ 𝑙𝑖𝑚 𝑁→∞ 𝑛=−𝑁 𝑁 𝑥[𝑛] 2 = 𝑛=−∞ ∞ 𝑥[𝑛] 2 Average power is: 𝑃 ∞ ≜ 𝑙𝑖𝑚 𝑁→∞ 1 2𝑁+1 𝑛=−𝑁 𝑁 𝑥[𝑛] 2

Signal Energy and Power For infinite time interval we have three classes: 𝑬 ∞ <∞, Therefore 𝑃 ∞ = lim 𝑇→∞ 𝐸 ∞ 2𝑇 =0 Example: A finite-energy signal is a signal that takes on the value 1 for [0,1] and 0 otherwise. In this case, 𝐸 ∞ = 1 and 𝑃 ∞ = 0.

Signal Energy and Power For infinite time interval: Signals with finite average power, that is : 𝑃 ∞ >0 Therefore 𝐸 ∞ =∞ This makes sense, because if the signal has a nonzero average power per unit time, then the summation of the average power over infinite time interval will yield infinite energy.

Signal Energy and Power For infinite time interval: Signals with finite average power (cont.) Example: 𝑥 𝑛 =4 𝑃 ∞ = lim 𝑁→∞ −𝑁 𝑁 𝑥[𝑛] 2 2𝑁+1 = lim 𝑁→∞ 16(2𝑁+1) (2𝑁+1) =16

Signal Energy and Power For infinite time interval: Neither 𝑷 ∞ nor 𝑬 ∞ is finite Example: 𝑥 𝑡 =𝑡 It is left to be checked by the reader that both 𝑷 ∞ and 𝑬 ∞ are infinite values.

TRANSFORMATIONS OF: THE INDEPENDENT VARIABLE In real life signals are transformed in a way that achieves a certain goal. Examples: In a high-fidelity audio system, an input signal representing music as recorded on a cassette or compact disc is modified in order to enhance desirable characteristics, to remove recording noise, or to balance the several components of the signal (e.g., treble and bass).

TRANSFORMATIONS OF: THE INDEPENDENT VARIABLE Examples (cont.): In an aircraft control system, signals corresponding to the actions of the pilot are transformed by electrical and mechanical systems into changes in aircraft thrust or the positions of aircraft control surfaces such as the rudder or ailerons, which in turn are transformed through the dynamics and kinematics of the vehicle into changes in aircraft velocity and heading

TRANSFORMATIONS OF: THE INDEPENDENT VARIABLE Time Shifting: the signal is shifted along the time axis to the right or to the left keeping the shape unchanged. Continuous-time functions For a signal 𝑔(𝑡)

TRANSFORMATIONS OF: THE INDEPENDENT VARIABLE Time Shifting-Continuous-time functions (cont.) If we have another 𝑓(𝑡), which is a shifted copy of 𝑔(𝑡) to the right with two time units. From the graph, 𝑓 0 =𝑔 −2 , 𝑓 1 =𝑔 −1 , 𝑓 2 = 𝑔(0), and so on. Therefore, 𝑓 𝑡 =𝑔 𝑡−2

TRANSFORMATIONS OF: THE INDEPENDENT VARIABLE Time Shifting-Continuous-time functions (cont.) Similarly, if 𝑓 𝑡 =𝑔 𝑡+2 , 𝑓(𝑡) is a shifted copy to the left of 𝑔 𝑡 .

TRANSFORMATIONS OF: THE INDEPENDENT VARIABLE Time Shifting-Discrete-time functions (cont.) For a discrete function 𝜁[𝑛], a shifted copy to the right is 𝜉 𝑛 =𝜁[𝑛−2] 𝜉 𝜁

TRANSFORMATIONS OF: THE INDEPENDENT VARIABLE Time Shifting-Discrete-time functions (cont.) For a discrete function 𝜁[𝑛], a shifted copy to the left is 𝜉 𝑛 =𝜁[𝑛+2] 𝜉 𝜁

TRANSFORMATIONS OF: THE INDEPENDENT VARIABLE Time Shifting For a continous-time signal 𝑥(𝑡): 𝑥 𝑡− 𝑡 0 represents a delayed version of 𝑥(𝑡). 𝑥(𝑡+ 𝑡 0 ) represents an advanced version of 𝑥(𝑡). For a discrete-time signal 𝑥[𝑛]: 𝑥[𝑛− 𝑛 0 ] represents a delayed version of 𝑥 𝑛 . 𝑥[𝑛+ 𝑛 0 ] represents an advanced version of 𝑥[𝑛].

TRANSFORMATIONS OF: THE INDEPENDENT VARIABLE Time Reversal (Continuous –Time) If we have a function 𝑥(𝑡), Then 𝑥(−𝑡)

TRANSFORMATIONS OF: THE INDEPENDENT VARIABLE Time Reversal (Discrete–Time) If we have a function 𝑥[𝑛], Then 𝑥[−𝑛]

TRANSFORMATIONS OF: THE INDEPENDENT VARIABLE Time Scaling For a signal 𝑥(𝑡), Slowed down version to the half speed is 𝑥( 𝑡 2 ) and speeded up version to the double speed is 𝑥(2𝑡) 𝑥(𝑡) 𝑥(2𝑡) 𝑥( 𝑡 2 )

TRANSFORMATIONS OF: THE INDEPENDENT VARIABLE The transforming of the independent variable of a given signal 𝑥(𝑡) to obtain a signal of the form 𝑥(𝛼𝑡 + 𝛽), where 𝛼 and 𝛽 are given numbers. Such a transformation of the independent variable preserves the shape of 𝑥(𝑡), except that: The resulting signal may be linearly stretched if 𝛼 <1, linearly compressed if 𝛼 >1, reversed in time if 𝛼 < 0, and shifted in time if 𝛽 is nonzero

TRANSFORMATIONS OF: THE INDEPENDENT VARIABLE Example : For a signal 𝑥 𝑡 : 𝑥(𝑡+1): The signal 𝑥(𝑡 + 1) corresponds to an advance (shift to the left) by one unit along the 𝑡 axis. We note that the value of 𝑥(𝑡) at 𝑡 = 𝑡𝑜 occurs in 𝑥(𝑡 + 1) at 𝑡 = 𝑡𝑜 − 1.

TRANSFORMATIONS OF: THE INDEPENDENT VARIABLE Example (cont.): For a signal 𝑥 𝑡+1 : 𝑥 −𝑡+1 : The signal 𝑥( − 𝑡 + 1), which may be obtained by replacing 𝑡 withtin 𝑥(𝑡 + 1). That is, 𝑥( − 𝑡 + 1) is the time reversed version of 𝑥(𝑡 + 1).

TRANSFORMATIONS OF: THE INDEPENDENT VARIABLE Example (cont.): For a signal 𝑥 𝑡 : 𝑥( 3 2 𝑡): The signal 𝑥( 3 2 𝑡) corresponds to a linear compression of 𝑥(𝑡) by a factor of 2 3 We note that .the value of 𝑥(𝑡) at 𝑡 = 𝑡𝑜 occurs in 𝑥( 3 2 𝑡) at 𝑡 = 2 3 𝑡𝑜

TRANSFORMATIONS OF: THE INDEPENDENT VARIABLE Example (cont.): For the signal 𝑥( 3 2 𝑡): 𝑥( 3 2 𝑡+1): The compressed signal is advanced (shifted to the left) by 1 unit .

TRANSFORMATIONS OF: THE INDEPENDENT VARIABLE In addition to their use in representing physical phenomena such as the time shift in a sonar signal and the speeding up or reversal of an audiotape, transformations of the independent variable are extremely useful in signal and system analysis. Transformations of the independent variable will be used to introduce and analyze the properties of systems. These transformations are also important in defining and examining some important properties of signals.

Thanks