- Data Structure
- Networking
- RDBMS
- Operating System
- Java
- MS Excel
- iOS
- HTML
- CSS
- Android
- Python
- C Programming
- C++
- C#
- MongoDB
- MySQL
- Javascript
- PHP
- Physics
- Chemistry
- Biology
- Mathematics
- English
- Economics
- Psychology
- Social Studies
- Fashion Studies
- Legal Studies
- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who
Properties of Discrete Time Unit Impulse Signal
What is a Discrete Time Impulse Sequence?
The discrete time unit impulse sequence 𝛿[𝑛], also called the unit sample sequence, is defined as,
$$\mathrm{\delta \left [ n \right ]=\left\{\begin{matrix} 1\; for\: n=0\ 0\; for \: n
eq 0\ \end{matrix}\right.}$$
Properties of Discrete Time Unit Impulse Sequence
Scaling Property
According to the scaling property of discrete time unit impulse sequence,
𝛿[𝑘𝑛] = 𝛿[𝑛]
Where, k is an integer.
Proof − By the definition of the discrete time unit impulse sequence,
$$\mathrm{\delta \left [ n \right ]=\left\{\begin{matrix} 1\; for\: n=0\ 0\; for \: n
eq 0\ \end{matrix}\right.}$$Similarly, for the scaled unit impulse sequence,
$$\mathrm{\delta \left [ kn \right ]=\left\{\begin{matrix} 1\; for\: kn=0\ 0\; for \: kn
eq 0\ \end{matrix}\right.}$$ $$\mathrm{\Rightarrow \delta \left [ kn \right ]=\left\{\begin{matrix} 1\; for\: n=\frac{0}{k}=0\ 0\; for \: n
eq\frac{0}{k}
eq 0\ \end{matrix}\right.=\left\{\begin{matrix} 1\; \; for\: n=0\ 0\; \; for\: n
eq 0\ \end{matrix}\right.=\delta \left [ n \right ]}$$Product Property
𝑥[𝑛]𝛿[𝑛 − 𝑛0] = 𝑥[𝑛0]𝛿[𝑛 − 𝑛0]
Proof − By the definition of the unit impulse signal, we know,
$$\mathrm{\delta \left [ n-n_{0} \right ]=\left\{\begin{matrix} 1\: \: for\: n=n_{0}\ 0\: \: for\: n
eq n_{0}\ \end{matrix}\right.}$$As from the expression it is clear that the impulse sequence has a non-zero value only at 𝑛 = 𝑛0. Therefore,
𝑥[𝑛]𝛿[𝑛 − 𝑛0] = 𝑥[𝑛0]𝛿[𝑛 − 𝑛0]
Shifting Property
$$\mathrm{x\left [ n \right ]=\sum_{k=-\infty }^{\infty }x\left [ k \right ]\delta \left [ n-k \right ]}$$Proof − By using the product property of the discrete time unit impulse sequence, we have,
𝑥[𝑛]𝛿[𝑛 − 𝑛0] = 𝑥[𝑛0]𝛿[𝑛 − 𝑛0] … (1)
Putting k in place of 𝑛0 in equation (1), we get,
𝑥[𝑛]𝛿[𝑛 − 𝑘] = 𝑥[𝑘]𝛿[𝑛 − 𝑘]
$$\mathrm{\Rightarrow \sum_{k=-\infty }^{\infty }x\left [ n \right ]\delta \left [ n-k \right ]=\sum_{k=-\infty }^{\infty }x\left [ k \right ]\delta \left [ n-k \right ]}$$ $$\mathrm{\Rightarrow x\left [ n \right ]\sum_{k=-\infty }^{\infty }\delta \left [ n-k \right ]=\sum_{k=-\infty }^{\infty }x\left [ k \right ]\delta \left [ n-k \right ]}$$ $$\mathrm{\because \sum_{k=-\infty }^{\infty }\delta \left [ n-k \right ]=1}$$ $$\mathrm{\therefore x\left [ n \right ]=\sum_{k=-\infty }^{\infty }x\left [ k \right ]\delta \left [ n-k \right ]}$$The discrete time unit impulse sequence is the first difference of discrete time unit step sequence. That is,
𝛿[𝑛] = 𝑢[𝑛] − 𝑢[𝑛 − 1]
Proof − By the definition of the discrete time unit step sequence,
$$\mathrm{u\left [ n \right ]=\sum_{k=0}^{\infty }\delta \left [ n-k \right ] =\delta \left [ n \right ]+\sum_{k=1}^{\infty }\delta \left [ n-k \right ] }$$ $$\mathrm{\because u\left [ n-1 \right ]=\sum_{k=1}^{\infty }\delta \left [ n-k \right ]}$$∴ 𝑢[𝑛] = 𝛿[𝑛] + 𝑢[𝑛 − 1]
⟹ 𝛿[𝑛] = 𝑢[𝑛] − 𝑢[𝑛 − 1]