Properties of Discrete Time Unit Impulse Signal



What is a Discrete Time Impulse Sequence?

The discrete time unit impulse sequence ?[?], 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,

    ?[??] = ?[?]

    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

    ?[?]?[? − ?0] = ?[?0]?[? − ?0]

    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 ? = ?0. Therefore,

    ?[?]?[? − ?0] = ?[?0]?[? − ?0]

  • 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,

    ?[?]?[? − ?0] = ?[?0]?[? − ?0]   … (1)

    Putting k in place of ?0 in equation (1), we get,

    ?[?]?[? − ?] = ?[?]?[? − ?]

    $$\mathrm{\Rightarrow \sum_{k=-\infty }^{\infty }x\left [ n \right ]\delta \left [ n-k \right ]=\sum_{k=-\infty }^{\infty }x\left [ k \right ]\delta \left [ n-k \right ]}$$ $$\mathrm{\Rightarrow x\left [ n \right ]\sum_{k=-\infty }^{\infty }\delta \left [ n-k \right ]=\sum_{k=-\infty }^{\infty }x\left [ k \right ]\delta \left [ n-k \right ]}$$ $$\mathrm{\because \sum_{k=-\infty }^{\infty }\delta \left [ n-k \right ]=1}$$ $$\mathrm{\therefore x\left [ n \right ]=\sum_{k=-\infty }^{\infty }x\left [ k \right ]\delta \left [ n-k \right ]}$$
  • The discrete time unit impulse sequence is the first difference of discrete time unit step sequence. That is,

    ?[?] = ?[?] − ?[? − 1]

    Proof − By the definition of the discrete time unit step sequence,

    $$\mathrm{u\left [ n \right ]=\sum_{k=0}^{\infty }\delta \left [ n-k \right ] =\delta \left [ n \right ]+\sum_{k=1}^{\infty }\delta \left [ n-k \right ] }$$ $$\mathrm{\because u\left [ n-1 \right ]=\sum_{k=1}^{\infty }\delta \left [ n-k \right ]}$$

    ∴ ?[?] = ?[?] + ?[? − 1]

    ? ?[?] = ?[?] − ?[? − 1]

Updated on: 2021-11-12T11:44:22+05:30

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