DISCRETE_SIGNALS - Discrete Time Signal Definition Notation Classification Manipulation Discrete Time Signal As discussed before a signal is a parameter

DISCRETE_SIGNALS - Discrete Time Signal Definition Notation...

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Discrete - Time Signal Definition Notation Classification Manipulation
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Discrete Time Signal As discussed before a signal is a parameter that depends on an independent parameter Definition : A Discrete-time Signal is a function of an independent variable that is an integer and is formally denoted by x = {x(n)} , - ∞ < n < ∞ This means that the independent parameter of the Discrete-time signal has to be represented by integers. For eg: if we sample an analog signal X(t) at time instants 0, T s , 2T s , …. i.e at t=nT s , then the discrete time signal obtained is denoted by x(n) and x(n) = X(nT s ) So here the independent parameter is time and n=0,1,2,3,… represent t=0, T s , 2T s , …. A discrete time signal is basically just a sequence of numbers, this will become clear as we discuss representation of discrete time signals
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Representation of Discrete Time Signals Graphical Representation A Discrete time signal may be represented graphically as shown in the figure. This is known as the lollipop representation. Note that the signal is represented only at integers, -9,-8,….,0,…. This is because discrete time signal domain is only integers. In between integers the signal is not equal to zero , In fact at these points (for eg between 1 and 2) , the signal does not exist !!! Functional Representation Another kind of representation is by denoting the discrete time signal as a function Eg: 1) x(n) = Asin(ω 0 n+θ) 3n-5, if -9≤n<0 2) x(n) = 4n+2, if 0 ≤n<5 0, otherwise
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Sequential Representation A discrete time signal may also be represented as a sequence of numbers. For example x = {2,3,4,9,3} is a signal. What this means is that n=0 is at the arrow, i.e. x(0)=3, and therefore x(-1)=2; x(1)=4; x(2)=9; x(3)=3. Therefore we see that any discrete time signal is basically just a sequence of numbers. It is for this reason we often refer to discrete time signals as sequences . Finite Duration Sequence: If a sequence is non-zero only over a finite period of time, e.g. x={1,4,1}; This sequence has 3 samples, so it is called a 3 point sequence. Infinite Duration Sequence: If a sequence is non-zero over an infinite duration of time, so it may range over (- ∞ <n< ∞) or (- ∞<n<a) or (a<n< ∞ ). Eg: {…….0,4,3,-2,1,6,7,……….} Representation of Discrete Time Signals contd.
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Some Basic/Important Sequences/Signals Finite duration sequences: Unit Sample Sequence: δ(n) = 1 , n=0 0 , otherwise Infinite duration sequences: Unit Step Sequence: u(n) = 1 , n=0 0 , otherwise Unit Ramp Sequence: u r (n) = n , n≥0 0 , otherwise Exponential Sequence: x(n)= a n , for all n Sinusoidal Sequence: x(n)= Asin(ω 0 n+θ) 0<a<1
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Classification of Discrete Time Signals Cannot predict value of the signal, varies randomly, hence also called random signal All past, present, future values of the signal are known. Can predict the value of the signal Non- Deterministic Deterministic Energy of the signal is infinite, power is finite, E ∞ and 0<P<∞ Eg: All periodic signals Energy of the signal is finite, 0<E<∞, All finite duration signals are energy signals Power Signal Energy Signal Does not have any N for which condition is satisfied If signal satisfies the
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