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Signals - Signals Signals are represented mathematically as...

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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 discrete-time signal when the time takes a discrete set of values. A discrete-time 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( ) ( 0 φ ϖ + = t A t x continuous time ) , ( -∞ t is a real number ) ˆ cos( ] [ 0 φ ϖ + = n A n x discrete time ) , ( -∞ n is an integer, ) , ( ˆ 0 π π ϖ - where A , 0 ϖ , and φ are called the amplitude, radian frequency, and phase shift of the sinusoid, respectively, while 0 ˆ ϖ is the so-called normalized radian frequency. Example 10 2 0 × = π ϖ rad/s, 0 = φ , A= 1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 t x(t) Example: 6 / ˆ 0 π ϖ = , 0 = φ , A= 1 { x [ n ]}={… 0.844, 0.866, 1, 0.866, 0.844 …} 1
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We use an arrow to identify x [0] at time n =0. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -1 -0.8 -0.6 -0.4 -0.2 0 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 0 ) 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 0 T which is the minimum value to satisfy the following relation 2
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] ) ( cos[ ) cos( 0 0 0 φ ϖ φ ϖ + + = + T t t This relation gives π ϖ 2 0 0 = T The frequency of a continuous time sinusoid is defined as the number of cycles per second. The period of the sinusoid, denoted by 0 T , is the time elapsed by one cycle of sinusoid. The relation of frequency to period is 0 0 1 T f = , unit = Hertz (sec -1 ) The frequency 0 f is related to radian frequency 0 ϖ as follows 0 0 0 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( ) ( 0 φ ϖ + = t A t x are located at Maxima: 0 max / ) 2 ( ϖ φ π - = k t Minima 0 min / ] ) 1 2 [( ϖ φ π - + = k t Example: 10 2 0 × = π ϖ rad/s, 3 / π φ = 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 time in sec amplitude 3
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033 . 0 ) 10 2 /( ) 3 / ( min = × - = π π π t , 0833 . 0 ) 10 2 /( ) 3 / 2 ( max = × - = π π π t Equivalently, the phase shift determines how much the maximum of the sinusoid is shifted with respect to t =0. The sinusoidal signal with a time shift 1 t is expressed as ) cos( )) ( cos( ) ( 0 1 0 1 φ ϖ ϖ + = - = - t A t t A t t x The time shift corresponding to the phase shift φ is φ ϖ - = 1 0 t or 0 1 ϖ φ - = t For the above sinusoidal example, s t 60 / 1 ) 10 2 /( 3 / 1 - = × - = π π Example: Illustration of time-shifting Complex exponentials and phasors
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