a) A system is said to be additive if the response to a sum of inputs is equal to the sum of the responses to the individual inputs. A system is said to be homogeneous if the response to a scaled input is equal to the scaled response to the original input.

If a system is additive, then the response to a zero input (which can be thought of as the sum of zero and zero) is the sum of the responses to two zero inputs. Since the response to a zero input must be the same each time, this implies that the response to a zero input is zero.

If a system is homogeneous, then the response to a zero input (which can be thought of as zero times some input) is zero times the response to that input. Since zero times anything is zero, this implies that the response to a zero input is zero.

b) Consider a system that squares its input. This system is not additive, because the response to the sum of two inputs (which is the square of the sum) is not generally equal to the sum of the responses to the individual inputs (which are the squares of the inputs). This system is not homogeneous, because the response to a scaled input (which is the square of the scaled input) is not generally equal to the scaled response to the original input (which is the square of the input). However, the response to a zero input is zero, because zero squared is zero.

c) From part a), we can conclude that if the input to a linear system is zero between times n1 and n2, then its output must also be zero between these same times. This is because a linear system is both additive and homogeneous, so it has the property that if the input is identically zero, then the output is also identically zero. Therefore, if the input is zero for a certain period of time, then the output must also be zero for that same period of time.

Question

a) A system is said to be additive if the response to a sum of inputs is equal to the sum of the responses to the individual inputs. A system is said to be homogeneous if the response to a scaled input is equal to the scaled response to the original input.

If a system is additive, then the response to a zero input (which can be thought of as the sum of zero and zero) is the sum of the responses to two zero inputs. Since the response to a zero input must be the same each time, this implies that the response to a zero input is zero.

If a system is homogeneous, then the response to a zero input (which can be thought of as zero times some input) is zero times the response to that input. Since zero times anything is zero, this implies that the response to a zero input is zero.

b) Consider a system that squares its input. This system is not additive, because the response to the sum of two inputs (which is the square of the sum) is not generally equal to the sum of the responses to the individual inputs (which are the squares of the inputs). This system is not homogeneous, because the response to a scaled input (which is the square of the scaled input) is not generally equal to the scaled response to the original input (which is the square of the input). However, the response to a zero input is zero, because zero squared is zero.

c) From part a), we can conclude that if the input to a linear system is zero between times n1 and n2, then its output must also be zero between these same times. This is because a linear system is both additive and homogeneous, so it has the property that if the input is identically zero, then the output is also identically zero. Therefore, if the input is zero for a certain period of time, then the output must also be zero for that same period of time.

Knowee AI · Accepted Answer

a) A system is said to be additive if the response to a sum of inputs is equal to the sum of the responses to the individual inputs. A system is said to be homogeneous if the response to a scaled input is equal to the scaled response to the original input.

If a system is additive, then the response to a zero input (which can be thought of as the sum of zero and zero) is the sum of the responses to two zero inputs. Since the response to a zero input must be the same each time, this implies that the response to a zero input is zero.

If a system is homogeneous, then the response to a zero input (which can be thought of as zero times some input) is zero times the response to that input. Since zero times anything is zero, this implies that the response to a zero input is zero.

b) Consider a system that squares its input. This system is not additive, because the response to the sum of two inputs (which is the square of the sum) is not generally equal to the sum of the responses to the individual inputs (which are the squares of the inputs). This system is not homogeneous, because the response to a scaled input (which is the square of the scaled input) is not generally equal to the scaled response to the original input (which is the square of the input). However, the response to a zero input is zero, because zero squared is zero.

c) From part a), we can conclude that if the input to a linear system is zero between times n1 and n2, then its output must also be zero between these same times. This is because a linear system is both additive and homogeneous, so it has the property that if the input is identically zero, then the output is also identically zero. Therefore, if the input is zero for a certain period of time, then the output must also be zero for that same period of time.

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