Signals and Systems Networks and Communication Department Chapter (1)

Lecture Contents Classifications of Signals(cont.) Sinusoidal Signals Transformations of the independent variable The Unit Impulse and Unit Step Function Signals and Systems Analysis

1.2 Classifications of Signals (Cont.) Signals and Systems Analysis

1.2 Classifications of Signals (Cont.) Sinusoidal Signals: Signals can be expressed as a function of time and a function of frequency. Time domain . Frequency domain . The frequency domain view of a signal is more important to an understanding of data transmission than a time domain view. Signals and Systems Analysis

1.2 Classifications of Signals (Cont.) Sinusoidal Signals: Analog signal signal intensity varies smoothly with no breaks Digital signal signal intensity maintains a constant level and then abruptly changes to another level Periodic signal signal pattern repeats over time Aperiodic signal signal pattern not repeated over time Time domain concepts Signals and Systems Analysis

1.2 Classifications of Signals (Cont.) Sinusoidal Signals: Time domain concepts Sine wave is the fundamental periodic signal. A general sine wave can be represented by three parameters : Peak amplitude (A) Frequency (f) Phase (φ) s(t) = A sin(2π f t +Φ) Signals and Systems Analysis

1.2 Classifications of Signals (Cont.) Sinusoidal Signals: Peak amplitude (A) maximum strength of signal typically measured in volts Frequency (f) rate at which the signal repeats (the number of times a signal makes a complete cycle within a given time frame) Hertz (Hz) or cycles per second period (T) is the amount of time for one repetition T = 1/f Phase (φ) relative position in time within a single period of signal Time domain concepts Signals and Systems Analysis

1.2 Classifications of Signals (Cont.) Sinusoidal Signals: Time domain concepts Signals and Systems Analysis

1.2 Classifications of Signals (Cont.) Sinusoidal Signals: Time domain concepts When a signal function is : s(t) = A sin(2π f t +Φ) Find the peak , frequency and phase of the following example: x(t)= 6 sin(2 4 t + /4) ?? Signals and Systems Analysis

1.2 Classifications of Signals (Cont.) Sinusoidal Signals: Varying sine waves : Time domain concepts Signals and Systems Analysis

1.2 Classifications of Signals (Cont.) Sinusoidal Signals: for a sinusoidal transmission, at a particular instant of time, the intensity of the signal varies in a sinusoidal way as a function of distance from the source. Time domain concepts the wavelength (λ) of a signal is the distance occupied by a single cycle can also be stated as the distance between two points of corresponding phase of two consecutive cycles assuming signal velocity v, then the wavelength is related to the period as λ = vT or equivalently λf = v especially when v=c •c = 3*108 ms-1 (speed of light in free space) Signals and Systems Analysis

1.2 Classifications of Signals (Cont.) Sinusoidal Signals: Frequency domain concepts Using discipline Known as Fourier analysis (any signal is made up of components at various frequencies, in which each component is a sinusoid ). By adding sinusoidal signals , any electromagnetic signals can constructed Any electromagnetic signal can be shown to consist of a collection of sine waves at different frequencies. Eg. s(t) = [(4/π) x (sin(2πft) + (1/3) sin(2π(3f)t)] the componets of this signal are sine waves of frequencies f an 3f Signals and Systems Analysis

1.2 Classifications of Signals (Cont.) Sinusoidal Signals: Frequency domain concepts Signals and Systems Analysis

1.2 Classifications of Signals (Cont.) Sinusoidal Signals: Frequency domain concepts Frequency, Spectrum and Bandwidth: Frequency is measured in Hertz (Hz), or cycles per second 𝑥(𝑡)= A sin(2f t + φ) e.g. S = 5sin (25t) here 𝑓=5Hz. Spectrum: Range of frequencies that a signal spans from minimum to maximum, e.g. 𝑥 𝑡 = 𝑥1 𝑡 + 𝑥2 𝑡 where 𝑥1 𝑡 =sin(25t) and 𝑥2 𝑡 =sin(27t). Here spectrum SP={5Hz,7Hz}. Bandwidth : Absolute value of the difference between the lowest and highest frequencies of a signal. In the above example bandwidth is BW=2Hz. Signals and Systems Analysis

1.2 Classifications of Signals (Cont.) Sinusoidal Signals: any transmission system has a limited band of frequencies this limits the data rate that can be carried on the transmission medium square waves have infinite components and hence an infinite bandwidth most energy in first few components limiting bandwidth creates distortions Frequency domain concepts Signals and Systems Analysis

1.3 Transformations of the independent variable (time) Signals and Systems Analysis

Transformations of time Time Reflection. Time Shiftting Time Scaling 17-Nov-18 Networks and Communication Department

1.3 Transformations of the independent variable: Reflection Signals and Systems Analysis

1.3 Transformations of the independent variable (Cont.): Shifting Signals and Systems Analysis

1.3 Transformations of the independent variable (Cont.): Scaling Signals and Systems Analysis

1.4 The Unit Impulse and Unit Step Function Signals and Systems Analysis

1.4 The Unit Impulse and Unit Step Function Signals and Systems Analysis

1.4 The Unit Impulse and Unit Step Function (Cont.) Definition of unit impulse: 𝑢 𝑡 = 0 𝑖𝑓 𝑡<0 1 𝑖𝑓 𝑡 ≥0 𝑢 t−𝑐 = 0 𝑖𝑓 𝑡<𝑐 1 𝑖𝑓 𝑡 ≥𝑐 𝐵𝑢 t−𝑐 = 0 𝑖𝑓 𝑡<𝑐 𝐵 𝑖𝑓 𝑡≥𝑐

1.4 The Unit Impulse and Unit Step Function (Cont.) 1.4.2 The Continuous-Time Unit Step: 𝑢 𝑡 = 0, 𝑡<0 1, 𝑡>0 Note: The unit step is discontinuous at 𝑡=0 and that value is undefined. Similarly, the shifted unit step function 𝑢(𝑡− 𝑡 0 ) is defined as 𝑢 𝑡− 𝑡 0 = 1 𝑡> 𝑡 0 0 𝑡< 𝑡 0 Signals and Systems Analysis

1.4 The Unit Impulse and Unit Step Function (Cont.) 1.4.1 The Discrete-Time Unit Step (Cont.) 𝑢 𝑛 = 0, 𝑛<0 1, 𝑛≥0 Note: that the value of u[n] at n = 0 is defined. The shifted unit step (or sample) sequence 𝑢 𝑛−𝑘 is defined as: 𝑢 𝑛−𝑘 = 0, 𝑛<𝑘 1, 𝑛≥𝑘 Note: that the value of 𝑢 𝑛 at n = 0 is defined. Signals and Systems Analysis

1.4 The Unit Impulse and Unit Step Function (Cont.) Unit Step examples : 3𝑢 t−2 = 0 𝑖𝑓 𝑡<2 3 𝑖𝑓 𝑡≥2 𝑢 𝑛+3 = 0, 𝑛<−3 1, 𝑛≥−3 Signals and Systems Analysis

1.4 The Unit Impulse and Unit Step Function (Cont.) Definition of unit impulse: 𝛿 𝑡 = +∞ 𝑖𝑓 𝑡=0 0 𝑖𝑓 𝑡≠0 𝛿 𝑡−𝑐 = +∞ 𝑖𝑓 𝑡=𝑐 0 𝑖𝑓 𝑡 ≠𝑐

1.4 The Unit Impulse and Unit Step Function (Cont.) 1.4.1 The Discrete-Time Unit Impulse 𝛿 𝑛 = 0, 𝑛≠0 1, 𝑛=0 The shifted unit impulse (or sample) sequence 𝛿 𝑛−𝑘 is defined as: 𝛿 𝑛−𝑘 = 1, 𝑛=𝑘 0, 𝑛≠𝑘 Signals and Systems Analysis

1.4 The Unit Impulse and Unit Step Function (Cont.) Signals and Systems Analysis

1.4 The Unit Impulse and Unit Step Function (Cont.) Unit impulse examples : 𝛿 𝑡+3 = +∞ 𝑖𝑓 𝑡=−3 0 𝑖𝑓 𝑡 ≠−3 𝛿 𝑛−2 = 1, 𝑛=2 0, 𝑛≠2 Signals and Systems Analysis

Summary Sinusoidal Signals. For each signal, there is a time domain function s(t) that specifies the amplitude of the signal at each instant in time . Similarly, there is a frequency domain function S(f) that specifies the peak amplitude of the constituent frequencies of the signal. Signals can be shift , scale or reflect on the time. For time shifting to signals there are two functions: Unit step. Unit impulse. Signals and Systems Analysis