What is the Z-transform relationship for a delayed sequence x[n-1]?

What is the Z-transform relationship for a delayed sequence x[n-1]?

• A. Z{x[n-1]} = z^{-1} X(z)
• B. Z{x[n-1]} = z X(z)
• C. Z{x[n-1]} = X(z)/z
• D. Z{x[n-1]} = z X(z^{-1})

Answer: (A)

Delaying a sequence by one sample shifts its index from n to n−1, and in the Z-domain that shift shows up as a factor z^{-1}. Specifically, Z{x[n−1]} = sum x[n−1] z^{-n} = sum x[k] z^{-(k+1)} = z^{-1} X(z). Since z^{-1} equals 1/z, you could also write this as X(z)/z, but the conventional form highlighting the delay is z^{-1} X(z).

Delaying a sequence by one sample shifts its index from n to n−1, and in the Z-domain that shift shows up as a factor z^{-1}. Specifically, Z{x[n−1]} = sum x[n−1] z^{-n} = sum x[k] z^{-(k+1)} = z^{-1} X(z). Since z^{-1} equals 1/z, you could also write this as X(z)/z, but the conventional form highlighting the delay is z^{-1} X(z).