This preview shows pages 1–5. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Signals Signals are represented mathematically as functions of one or more independent variables. A signal is called as continuous time signal when it is a function of continuous time. It is a discretetime signal when the time takes a discrete set of values. A discretetime signal is called as digital signal when its amplitude has discrete values. Sinusoids or sinusoidal signals or cosine signals or sine signals is a special class of signals. The general mathematical formula for a sinusoid is ) cos( ) ( φ ϖ + = t A t x continuous time ) , ( ∞∞ ∈ t is a real number ) ˆ cos( ] [ φ ϖ + = n A n x discrete time ) , ( ∞∞ ∈ n is an integer, ) , ( ˆ π π ϖ ∈ where A , ϖ , and φ are called the amplitude, radian frequency, and phase shift of the sinusoid, respectively, while ˆ ϖ is the socalled normalized radian frequency. Example 10 2 × = π ϖ rad/s, = φ , A= 1 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.210.80.60.40.2 0.2 0.4 0.6 0.8 1 t x(t) Example: 6 / ˆ π ϖ = , = φ , A= 1 { x [ n ]}={… 0.844, 0.866, 1, 0.866, 0.844 …} 1 We use an arrow to identify x [0] at time n =0. 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.210.80.60.40.2 0.2 0.4 0.6 0.8 1 n x[n] Properties of sine and cosine functions Property Equation Equivalence ) 2 / cos( sin π θ θ = or ) 2 / sin( cos π θ θ = ) 2 / cos( sin π θ θ + = , ) 2 / sin( cos π θ θ + = Periodicity θ π θ cos ) 2 cos( = + k , when k is an integer Evenness θ θ cos ) cos( = Oddness θ θ sin ) sin( = Zeros ) sin( = k π , when k is an integer Ones 1 ) 2 cos( = k π , when k is an integer Minus ones 1 )] ( 2 cos[ 2 1 = + k π , when k is an integer Basic Trigonometric Identities Identity Equation 1 1 cos sin 2 2 = + θ θ 2 θ θ θ 2 2 sin cos 2 cos = 3 θ θ θ cos sin 2 2 sin = 4 β α β α β α sin cos cos sin ) sin( ± = ± 5 β α β α β α sin sin cos cos ) cos( = ± Relationship between Frequency and Period Sinusoidal signal is a periodic signal having a period T which is the minimum value to satisfy the following relation 2 ] ) ( cos[ ) cos( φ ϖ φ ϖ + + = + T t t This relation gives π ϖ 2 = T The frequency of a continuous time sinusoid is defined as the number of cycles per second. The period of the sinusoid, denoted by T , is the time elapsed by one cycle of sinusoid. The relation of frequency to period is 1 T f = , unit = Hertz (sec1 ) The frequency f is related to radian frequency ϖ as follows 2 2 T f π π ϖ = = Relationship between phase shift and time shift The phase shift φ together with the frequency determines the time locations of the maxima and minima of a sinusoid. For given φ , the maxima and minima of ) cos( ) ( φ ϖ + = t A t x are located at Maxima: max / ) 2 ( ϖ φ π = k t Minima min / ] ) 1 2 [( ϖ φ π + = k t Example: 10 2 × = π ϖ rad/s, 3 / π φ = 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.210.80.60.40.2 0.2 0.4 0.6 0.8 1 time in sec amplitude 3 033 . ) 10 2 /( ) 3 / ( min = ×...
View
Full
Document
This note was uploaded on 01/11/2011 for the course EE 4015 taught by Professor Shuhungleung during the Fall '10 term at City University of Hong Kong.
 Fall '10
 ShuHungLeung
 Digital Signal Processing, Signal Processing

Click to edit the document details