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Discrete-Time Signals: Discrete-Time Signals:
Discrete-Time Signals: Discrete-Time Signals:
Time-Domain Representation Time-Domain Representation
Time-Domain Representation Time-Domain Representation
Signals represented as sequences of Discrete-time signal may also be written as
numbers, called samples a sequence of numbers inside braces:
Sample value of a typical signal or sequence {x[n]}={K,−0.2,2.2,1.1,0.2,−3.7,2.9,K}
denoted as x[n] with n being an integer in ↑
the range −∞≤n≤∞ In the above, x[−1]= −0.2, x[0]= 2.2, x[1]=1.1,
x[n] defined only for integer values of n and etc.
undefined for noninteger values of n The arrow is placed under the sample at
Discrete-time signal represented by {x[n]} time index n = 0
1 Copyright © 2005, S. K. Mitra 2 Copyright © 2005, S. K. Mitra
Discrete-Time Signals:
Discrete-Time Signals: Discrete-Time Signals:
Discrete-Time Signals: Time-Domain Representation
Time-Domain Representation Time-Domain Representation
Time-Domain Representation In some applications, a discrete-time
Graphical representation of a discrete-time sequence {x[n]} may be generated by
signal with real-valued samples is as shown periodically sampling a continuous-time
below: x (t)
signal at uniform intervals of time
a
3 Copyright © 2005, S. K. Mitra 4 Copyright © 2005, S. K. Mitra
Discrete-Time Signals:
Discrete-Time Signals: Discrete-Time Signals:
Discrete-Time Signals:
Time-Domain Representation
Time-Domain Representation Time-Domain Representation
Time-Domain Representation
Here, n-th sample is given by Unit of sampling frequency is cycles per
x[n]= x (t) =x (nT), n=K,−2,−1,0,1,K second, or hertz (Hz), if T is in seconds
a t=nT a
The spacing T between two consecutive Whether or not the sequence {x[n]} has
samples is called the sampling interval or been obtained by sampling, the quantity
sampling period x[n] is called the n-th sample of the
Reciprocal of sampling interval T, denoted sequence
as , is called the sampling frequency: {x[n]} is a real sequence, if the n-th sample
F
T x[n] is real for all values of n
F =1
T T Otherwise, {x[n]} is a complex sequence
5 Copyright © 2005, S. K. Mitra 6 Copyright © 2005, S. K. Mitra
Discrete-Time Signals: Discrete-Time Signals:
Discrete-Time Signals: Discrete-Time Signals:
Time-Domain Representation Time-Domain Representation
Time-Domain Representation Time-Domain Representation
A complex sequence {x[n]} can be written Example-{x[n]}={cos0.25n} is a real
as where sequence
{x[n]}={x [n]}+ j{x [n]}
re im j0.3n
x [n] x [n] {y[n]}={e }
and are the real and imaginary is a complex sequence
re im
parts of x[n] We can write
The complex conjugate sequence of {x[n]} {y[n]}={cos0.3n+ jsin0.3n}
is given by {x*[n]}={x [n]}− j{x [n]}
re im ={cos0.3n}+ j{sin0.3n}
Often the braces are ignored to denote a where {y [n]}={cos0.3n}
sequence if there is no ambiguity re
{y [n]}={sin0.3n}
7 8 im
Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra
Discrete-Time Signals:
Discrete-Time Signals: Discrete-Time Signals:
Discrete-Time Signals: Time-Domain Representation
Time-Domain Representation Time-Domain Representation
Time-Domain Representation Two types of discrete-time signals:
Example- - Sampled-data signals in which samples
{w[n]}={cos0.3n}− j{sin0.3n}={e−j0.3n} are continuous-valued
is the complex conjugate sequence of {y[n]} - Digital signals in which samples are
That is, discrete-valued
{w[n]}={y*[n]} Signals in a practical digital signal
processing system are digital signals
obtained by quantizing the sample values
either by rounding or truncation
9 Copyright © 2005, S. K. Mitra 10 Copyright © 2005, S. K. Mitra
Discrete-Time Signals: Discrete-Time Signals:
Discrete-Time Signals: Discrete-Time Signals:
Time-Domain Representation Time-Domain Representation
Time-Domain Representation Time-Domain Representation
Example- A discrete-time signal may be a finite-
length or an infinite-length sequence
plitude plitude Finite-length (also called finite-duration or
m m
A A finite-extent) sequence is defined only for a
time, t time, t finite time interval: N ≤n≤N
1 2
where and with
−∞ N2 Given by the norm of the signal
Lp-norm
N2 1/ p
n x =⎛ ∞ x[n]p⎞
p ⎜ ∑ ⎟
A left-sided sequence ⎝n=−∞ ⎠
N ≤0, where p is a positive integer
If a left-sided sequence is called a
2
anti-causal sequence
17 Copyright © 2005, S. K. Mitra 18 Copyright © 2005, S. K. Mitra
Discrete-Time Signals: Discrete-Time Signals:
Discrete-Time Signals: Discrete-Time Signals:
Time-Domain Representation Time-Domain Representation
Time-Domain Representation Time-Domain Representation
∞ L -norm x
The value of p is typically 1 or 2 or 1 1
is the mean absolute value of {x[n]}
L2-norm L -norm x
x ∞ ∞
2 is the peak absolute value of {x[n]}, i.e.
is the root-mean-squared (rms) value of
{x[n]} x ∞ = xmax
19 Copyright © 2005, S. K. Mitra 20 Copyright © 2005, S. K. Mitra
Discrete-Time Signals:
Discrete-Time Signals:
Operations on Sequences
Time-Domain Representation Operations on Sequences
Time-Domain Representation
Example A single-input, single-output discrete-time
Let{y[n]},0≤n≤ N −1, be an approximation of system operates on a sequence, called the
{x[n]}, 0 ≤ n ≤ N −1 input sequence, according some prescribed
An estimate of the relative error is given by the rules and develops another sequence, called
ratio of the L2-norm of the difference signal and the output sequence, with more desirable
the L2-norm of {x[n]}:
N−1 1/ p properties
⎛ y[n]−x[n]2⎞
⎜ ∑ ⎟
Erel =⎜n=0 ⎟ x[n] Discrete-time y[n]
⎜ N−1 2 ⎟ system
⎜ ∑ x[n] ⎟ Input sequence Output sequence
21 ⎝ n=0 ⎠ 22
Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra
Basic Operations
Operations on Sequences Basic Operations
Operations on Sequences
Product (modulation) operation:
For example, the input may be a signal x[n] × y[n]
corrupted with additive noise – Modulator y[n]= x[n]⋅w[n]
Discrete-time system is designed to w[n]
generate an output by removing the noise An application is in forming a finite-length
component from the input sequence from an infinite-length sequence
In most cases, the operation defining a by multiplying the latter with a finite-length
particular discrete-time system is composed sequence called an window sequence
of some basic operations Process called windowing
23 Copyright © 2005, S. K. Mitra 24 Copyright © 2005, S. K. Mitra
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