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 $(4.140)$.

  • $\begingroup$ I do not agree. The inputs and outputs of two figures are not consistent. $\endgroup$ Dec 17 '20 at 15:10
  • $\begingroup$ @DSPinfinity what do you mean exactly with not consistent? The change of notation? $\endgroup$
    – Gilles
    Dec 17 '20 at 15:17
  • $\begingroup$ 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. $\endgroup$ Dec 17 '20 at 15:28
  • $\begingroup$ @DSPinfinity, I've added an edit. Please consider visiting section 4.8.4 of the book. $\endgroup$
    – Gilles
    Dec 17 '20 at 15:39
  • $\begingroup$ I understand in terms of notation but since Fig 4.62 and 4.64 are parts in Fig 4.47, it makes very confusing and hence everything including notation must be consistent. $\endgroup$ Dec 17 '20 at 15:45

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