Characters used in exxpressions for the unit impulse in continuous time and discrete time:
Continuous Time Expression:
- ( \delta(t) ): This is the symbol used to represent the unit impulse function in continuous time.
- ( t ): This variable represents time, and it is the independent variable of the function.
- ( \infty ): This symbol represents infinity. In the context of the expression, it indicates that the function’s value is infinite at ( t = 0 ).
- ( 0 ): This value represents zero. The function’s value is zero for all ( t ) except ( t = 0 ).
- ( dt ): This represents an infinitesimally small increment in time. It’s used when integrating to calculate the area under the impulse function.
Discrete Time Expression:
- ( \delta[n] ): This is the symbol used to represent the unit impulse function in discrete time.
- ( n ): This variable represents the index or sample number in discrete time. It’s the independent variable of the function.
- ( 1 ): This value indicates that the unit impulse function has a value of 1 at ( n = 0 ), and it’s zero for all other ( n ).
- ( 0 ): This value represents zero. The function’s value is zero for all ( n ) except ( n = 0 ).
- ( \sum ): This symbol represents a summation, and it’s used to add up the values of the unit impulse function over all possible values of ( n ).
Overall, the expressions for the unit impulse in both continuous time and discrete time are mathematical representations that describe a pulse-like function that has a specific value (either infinite or 1) at a specific point (either ( t = 0 ) or ( n = 0 )) and is zero everywhere else. These expressions are foundational in signal processing and provide a way to mathematically analyze and describe the behavior of signals and systems.