Lecture 2-done.pdf

Remote Sensing Laboratory Dept. of Information Engineering and Computer Science University of Trento Via Sommarive, 14, I-38123 Povo, Trento, Italy

Digital Signal Processing Lecture 2

Begüm Demir

E-mail: [email protected] Web page: http://rslab.disi.unitn.it

Discrete Time Signals
A discrete time signal is a sequence of numbers


The length N of the signal is the total number of samples in the sequence.

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Operations on Sequences
Addition/Subtraction

x3[n]=x1[n]+x2[n] University of Trento, Italy

x4[n]=x1[n]-x2[n] B. Demir

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Operations on Sequences
Multiplication

x5[n]=x1[n]x2[n]


How will you apply multiplication if the sizes are not the same? University of Trento, Italy

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Operations on Sequences
Multiplication & Addition with constants

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Operations on Sequences
Shifting

y[n]=x[n-n0]  If n0 >0 time delay

advance University of Trento, Italy

If n0 <0 time advance

delay B. Demir

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Operations on Sequences
Time Reverse: y[n]= x[-n]

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Operations on Sequences y[n ] = x[α n ]

Signal compression

n y[ n ] = x   α 

Signal stretching

x[n ] is given below. Estimate x[2n ] and x[n / 2]

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Linear Interpolation

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Elementary Digital Signals
Unit Impulse (Digital Impulse, Discrete Time Impulse):

1 n = 0 δ [n ] =  0 n ≠ 0

1 n = m δ [n − m ] =  0 n ≠ m

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Exercises
Write the signals given below in terms of


if

δ [n ]

x[n] = δ (n) + δ (n − 1) + δ (n − 2) + 4δ (n − 3) + 5δ (n − 4) x[−n + 2] = ?

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Elementary Digital Signals
Unit Step Sequence:

1 n ≥ 0 u[n ] =  0 n < 0


Unit Impulse and Unit Step Sequences are related as follows:

δ [n ] = u[n ] − u[n − 1]

∞

u[n ] = ∑ δ ( n − k ) k =0

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Exercises

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Elementary Digital Signals
Real exponential sequence:

x[ n ] = a n

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Elementary Digital Signals
Real exponential sequence:

x(n) = a n

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Elementary Digital Signals
Sinusoidal sequence:

e jΩn + e − jΩn cos(Ωn ) = 2 e jΩn − e − jΩn sin(Ωn ) = 2j

x[n ] = A cos(Ωn + θ )

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Elementary Digital Signals
Complex exponential sequence:

x[n ] = e jΩn

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z = re jθ ( r = 1,θ = Ωn )

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Elementary Digital Signals
Complex exponential sequence:

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Properties of Sinusoidal/Complex Signals

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Properties of Sinusoidal Signals cos[(Ω + 2πr )n ] = cos(Ωn)

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Periodicity of Discrete Time Signals

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Examples π  Is x [ n ] = cos  n  periodic?  11 

Is x [ n ] = 5e j 4π n / 7 periodic?

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Examples
Let us assume that we have 3 sinusoidal signals x1[n], x2[n] and x3[n] that are periodic with N1, N2 and N3, respectively. The signal x4[n] given below is periodic?

x4 [ n] = ax1 [ n] + bx2 [ n] + cx3 [ n] ; x4 [ n] is periodic?

N 4 = LCM ( N1 , N 2 , N 3 )

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