# Is this an error in Oppenheim and Schafer's Discrete-Time Signal Processing?

In Discrete-Time Signal Processing by Alan V. Oppenheim and Ronald W. Schafer (3rd Ed.), in Figure 4.47 the input of D/A converter is $$\hat{y}[n]$$ but later in Figure 4.64 the input of D/A converter is $$\hat{x}[n]$$. Is this a mistake?

Normally, based on Figure 4.47 $$\hat{y}[n]$$ is the output of the discrete-time system with input $$\hat{x}[n]$$.

It is not a mistake. In section 4.8.4 the authors introduce the reconstruction of $$x_r(t)$$ from $$x[n]$$ in equation $$(4.140)$$; and in the paragraph below, the authors consider the input sequence and output signal to the ideal D/C converter as $$x[n]$$ and $$x_r(t)$$ respectively, as shown below $$x[n]{\longrightarrow}\boxed{\textit{Ideal D/C converter}}{\longrightarrow}x_r(t)$$ Which will be more in line with the last input-output block of Figure 4.47(a) shown as $$\cdots \quad y[n]{\longrightarrow}\boxed{\textit{D/C}}{\longrightarrow}y_r(t)$$
To clarify, the extension from ideal D/C converter to the D/A and their corresponding maths with inputs-outputs relations and the figures, please consider reading the full section 4.8.4 with emphasis on the change of notation just after equation $$(4.140)$$.
• Ie, the inputs and outputs of two figures are not consistent. $\hat{x}[n]$ in Fig. 4.47 is input to the discrete-time system but in Fig.4.62 and 4.64 it is the same signal in D/A. $\hat{x}[n]$ is processed by the discrete-time system and hence it will change unless the discrete-time system is unit gain. Dec 17 '20 at 15:28