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You're interpretation that a[u] and b[u] are both N length sequences is incorrect. a[u] is of length N because it is effectively truncated by the length of your input sequence. The period of b[u] is not necessarily N. Depending on the value of N, b[u] may not even be periodic, so what you have in Equation 7 is a linear convolution of two sequences. One sequence (a[u] has length N, the other (b[u]) has length unknown. The limits of summation in the convolution equation are strictly set by the length of $a[u]$ and don't imply circular convolution or anything about the length of sequence b[u].

Next you have to consider what range of values of b[u] are required to compute the convolution. Straight from the form of EQUATION 1, you see that b[u] in general must be defined in the range -[N-1] to N-1. The negative boundary is set by the most negative b[u] value required to calculate X[0] (i.e. you need b[0−(N-1)] for this calculation). The positive boundary is set by the maximum spectral step you want to evaluate in the chirp Z transform (X[K] max). From the way the question has been posed, X[k] is best interpreted as $X[N-1]$.

The reason your computations work with a range that only goes back as far as -[M-1] is because you started with an M length sequence and zero padded it up to length N which zeros out any b[u] values going back further than -[M-1].

You're interpretation that a[u] and b[u] are both N length sequences is incorrect. a[u] is of length N because it is effectively truncated by the length of your input sequence. The period of b[u] is not necessarily N. Depending on the value of N, b[u] may not even be periodic, so what you have in Equation 7 is a linear convolution of two sequences. One sequence (a[u] has length N, the other (b[u]) has length unknown. The limits of summation in the convolution equation are strictly set by the length of $a[u]$ and don't imply circular convolution or anything about the length of sequence b[u].

Next you have to consider what range of values of b[u] are required to compute the convolution. Straight from the form of EQUATION 1, you see that b[u] in general must be defined in the range -[N-1] to N-1. The negative boundary is set by the most negative b[u] value required to calculate X[0] (i.e. you need b[0−(N-1)] for this calculation). The positive boundary is set by the maximum spectral step you want to evaluate in the chirp Z transform (X[K] max). From the way the question has been posed, X[k] is best interpreted as $X[N-1]$.

The reason your computations work with a range that only goes back as far as -[M-1] is because you started with an M length sequence and zero padded it up to length N which zeros out any b[u] values going back further than -[M-1].

There is an interesting write up of Chirp-Z along with Goertzel's algorithm that I think puts this in proper perspective: http://www.google.com/url?sa=t&rct=j&q=goertzel%20algorithm&source=web&cd=3&sqi=2&ved=0CEIQFjAC&url=http%3A%2F%2Focw.mit.edu%2Fcourses%2Felectrical-engineering-and-computer-science%2F6-341-discrete-time-signal-processing-fall-2005%2Flecture-notes%2Flec20.pdf&ei=rawjUd6sM-LE0QHJ2IGQAQ&usg=AFQjCNEFEo7R7VSmk5jWYon0f4WqZYpvzg&bvm=bv.42553238,d.dmQ

You're interpretation that a[u] and b[u] are both N length sequences is incorrect. a[u] is of length N because it is effectively truncated by the length of your input sequence. The period of b[u] is not necessarily N. Depending on the value of N, b[u] may not even be periodic, so what you have in EQUATION 1 is a linear convolution of two sequences. One sequence (a[u] has length N, the other (b[u]) has length unknown. The limits of summation in the convolution equation are strictly set by the length of $a[u]$ and don't imply circular convolution or anything about the length of sequence b[u].

Next you have to consider what range of values of b[u] are required to compute the convolution. Straight from the form of EQUATION 1, you see that b[u] in general must be defined in the range -[N-1] to N-1. The negative boundary is set by the most negative b[u] value required to calculate X[0] (i.e. you need b[0−(N-1)] for this calculation). The positive boundary is set by the maximum spectral step you want to evaluate in the chirp Z transform (X[K] max). From the way the question has been posed, X[k] is best interpreted as $X[N-1]$.

The reason your computations work with a range that only goes back as far as -[M-1] is because you started with an M length sequence and zero padded it up to length N which zeros out any b[u] values going back further than -[M-1].

You're interpretation that a[u] and b[u] are both N length sequences is incorrect. a[u] is of length N because it is effectively truncated by the length of your input sequence. The period of b[u] is not necessarily N. Depending on the value of N, b[u] may not even be periodic, so what you have in Equation 7 is a linear convolution of two sequences. One sequence (a[u] has length N, the other (b[u]) has length unknown. The limits of summation in the convolution equation are strictly set by the length of $a[u]$ and don't imply circular convolution or anything about the length of sequence b[u].

Next you have to consider what range of values of b[u] are required to compute the convolution. Straight from the form of EQUATION 1, you see that b[u] in general must be defined in the range -[N-1] to N-1. The negative boundary is set by the most negative b[u] value required to calculate X[0] (i.e. you need b[0−(N-1)] for this calculation). The positive boundary is set by the maximum spectral step you want to evaluate in the chirp Z transform (X[K] max). From the way the question has been posed, X[k] is best interpreted as $X[N-1]$.

The reason your computations work with a range that only goes back as far as -[M-1] is because you started with an M length sequence and zero padded it up to length N which zeros out any b[u] values going back further than -[M-1].

You're interpretation that a[u] and b[u] are both N length sequences is incorrect. a[u] is of length N because it is effectively truncated by the length of your input sequence. The period of b[u] is not necessarily N. Depending on the value of N, b[u] may not even be periodic, so what you have in EQUATION 1 is a linear convolution of two sequences. One sequence (a[u] has length N, the other (b[u]) has length unknown. The limits of summation in the convolution equation are strictly set by the length of a[u] and don't imply circular convolution or anything about the length of sequence b[u].

Next you have to consider what range of values of b[u] are required to compute the convolution. Straight from the form of EQUATION 1, you see that b[u] in general must be defined in the range -[N-1] to N-1. The negative boundary is set by the most negative b[u] value required to calculate X[0] (i.e. you need b[0−(N-1)] for this calculation). The positive boundary is set by the maximum spectral step you want to evaluate in the chirp Z transform (X[K] max). From the way the question has been posed, X[k] is best interpreted as X[N-1].

The reason your computations work with a range that only goes back as far as -[M-1] is because you started with an M length sequence and zero padded it up to length N which zeros out any b[u] values going back further than -[M-1].

You're interpretation that a[u] and b[u] are both N length sequences is incorrect. a[u] is of length N because it is effectively truncated by the length of your input sequence. The period of b[u] is not necessarily N. Depending on the value of N, b[u] may not even be periodic, so what you have in EQUATION 1 is a linear convolution of two sequences. One sequence (a[u] has length N, the other (b[u]) has length unknown. The limits of summation in the convolution equation are strictly set by the length of $a[u]$ and don't imply circular convolution or anything about the length of sequence b[u].

Next you have to consider what range of values of b[u] are required to compute the convolution. Straight from the form of EQUATION 1, you see that b[u] in general must be defined in the range -[N-1] to N-1. The negative boundary is set by the most negative b[u] value required to calculate X[0] (i.e. you need b[0−(N-1)] for this calculation). The positive boundary is set by the maximum spectral step you want to evaluate in the chirp Z transform (X[K] max). From the way the question has been posed, X[k] is best interpreted as $X[N-1]$.

The reason your computations work with a range that only goes back as far as -[M-1] is because you started with an M length sequence and zero padded it up to length N which zeros out any b[u] values going back further than -[M-1].