This document discusses discrete-time signals and digital signal processing. It begins by defining discrete-time signals as sequences of numbers represented as x[n], where n is an integer. These sequences arise from periodically sampling analog signals. Basic operations on sequences like addition, multiplication, and delay are introduced. Discrete-time systems are classified as memoryless if the output only depends on the current input, or time-invariant if a delayed input results in a delayed output. Popular orthogonal bases for representing sequences are discussed, including the Fourier series and discrete cosine transform (DCT). Sampling and reconstruction of discrete-time signals from analog signals is also covered.
2. COLLEGE NAME - BENGAL INSTITUTE OF
TECHNOLOGY AND MANAGEMENT
STUDENT NAME - ANIRBAN BHOWMIK
DEPARTMENT - ECE
ROLL NO - 16300322051
SEM - 5th
3. Introduction
Discreet-Time signals are represented
mathematically as sequences of numbers
The sequence is denoted 𝑥[𝑛], and it is
written formally as
𝑥 = 𝑥 𝑛 ; −∞ < 𝑛 < ∞
where n is an integer number
In practice sequences arises from the
periodic sampling of an analog signal
3
4. Discrete-Time signals:
sequences
In this case the numeric value of the nth
number in the sequence is equal to the
value of the analog signal, 𝑥𝑎
(𝑡), at time
𝑛𝑇
𝑥𝑛 = 𝑥𝑎
[𝑛𝑇]
4
6. Basic sequences and sequence
operation
The product and sum of two sequences
x[n] and 𝑦[𝑛] are defined as the
sample by sample product and sum
Multiplication of a sequence 𝑥[𝑛] by a
number 𝛼 is defined as the
multiplication of each sample value by
𝛼
A sample 𝑦[𝑛] is said to be delayed or
shifted version of 𝑥[𝑛] if 𝑦 𝑛 = 𝑥[
𝑛 −
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7. Discrete time system
classifications
Systems can be classifieds into one of the
following categories
1. Memoryless Systems. A system is
classified into memoryless system if
the output 𝑦 𝑛 at every value of 𝑛
depends only on the input of 𝑥[𝑛] at
the same value of 𝑛. An example of a
memoryless system is the squarer
system described by 𝑦 𝑛 = 𝑥[𝑛] 2
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8. Discrete time system
classifications
Example show that the accumulator system
𝑦 𝑛 𝑘=−∞
= ∑
𝑛 𝑥[𝑘] is a time invariant system
solution
Assume that the input to the accumulator is
𝑥1
𝑛 = 𝑥[𝑛 − 𝑛0
], then its output is 𝑦1
𝑛 =
𝑘=−∞
∑
𝑛 𝑘=−∞
𝑥1
[𝑘] =
∑𝑛
𝑥[𝑘 − 𝑛0
]
Let 𝑘1
= 𝑘 − 𝑛0
This means that
𝑦1
𝑛 𝑘=−∞
= ∑
𝑛−𝑛0 𝑥[𝑘1
] = y[n − 𝑛0
]
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10. Representation of sequences
by Fourier transforms
𝑋 𝑒
𝑗𝜔 ∑
𝑛=−∞
In order to represent a given sequence by its
Fourier transform we can use the following
equation
∞
However the inverse Fourier transform is given
by
𝑥 𝑛 𝑒
−𝑗𝜔𝑛
𝑥 𝑛 =
1
2𝜋
∫
−𝜋
𝜋
𝑋 𝑒
𝑗𝜔
𝑒
𝑗𝜔𝑛
𝑑𝜔
1
0
11. Representation of sequence
Fourier transforms
For the discrete time signals, the value of
𝜔 is restricted to an interval of 2𝜋
The low frequency component of
discrete time signals are located around
𝜔 = 0
The high frequency component
are located around 𝜔 = ±𝜋
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12. Discrete Consine Transform
● Discrete Cosine Transform (DCT) has emerged as the
image transformation in most visual systems. DCT has
been widely deployed by modern video coding
standards, for example, MPEG, JVT etc.
● It is the same family as the Fourier Transform
➢ Converts data to frequency domain
● Represents data via summation of variable frequency
cosine waves.
● Captures only real components of the function.
➢ Discrete Sine Transform (DST) captures odd
(imaginary) components → not as useful.
➢ Discrete Fourier Transform (DFT) captures both
odd and even components → computationally
intense.
