The top part of Figure 8.1 shows the DCT $F(u)$ for a 1-D signal $f(i)$.
i. What does $F(5)$ tell you in terms of the image content in $f(i)$ ?
ii. What is the reason that $F(5)$ is less than 0 ?
(b) The Inverse Discrete Cosine Transform (IDCT) can be viewed as a reconstruction process in which the Cosine basis functions are scaled and added one by one to reconstruct $\tilde{f}(i)$. Assume $F(0)=500$ and $F(1)=-70$; in the space provided at the bottom part of the figure,
i. draw $\tilde{f}(i)$ after only the $F(0)$ is used,
ii. draw $\tilde{f}(i)$ after $F(0)$ and $F(1)$ are used.
Show all details including magnitudes, and explain why they look like that.

Question

The top part of Figure 8.1 shows the DCT $F(u)$ for a 1-D signal $f(i)$. 
i. What does $F(5)$ tell you in terms of the image content in $f(i)$ ?
ii. What is the reason that $F(5)$ is less than 0 ?
(b) The Inverse Discrete Cosine Transform (IDCT) can be viewed as a reconstruction process in which the Cosine basis functions are scaled and added one by one to reconstruct $\tilde{f}(i)$. Assume $F(0)=500$ and $F(1)=-70$; in the space provided at the bottom part of the figure,
i. draw $\tilde{f}(i)$ after only the $F(0)$ is used,
ii. draw $\tilde{f}(i)$ after $F(0)$ and $F(1)$ are used.
Show all details including magnitudes, and explain why they look like that.

Quizplus · Accepted Answer

The Answer is i.

F(5)

tells us about the contribution of the 5th cosine basis function to the signal

f(i)

. It indicates the presence of high-frequency content in the signal, as the higher frequency components tend to have smaller magnitudes in the DCT coefficients.

ii. The reason

F(5)

is less than 0 is because the 5th cosine basis function has a negative contribution to the signal

f(i

. This means that the signal contains information that is out of phase with this particular cosine basis function, resulting in a negative coefficient.

(b) i. After using

F(0)

to reconstruct

ilde{f}(i)

, the signal will be a constant value equal to the magnitude of

F(0)

, which is 500.

ii. After using both

F(0)

and

F(1)

to reconstruct

ilde{f}(i)

, the signal will have a constant component from

F(0)

and a sinusoidal component from

F(1)

. The magnitude of the sinusoidal component will be the magnitude of

F(1)

, which is 70, and the shape of the sinusoid will be determined by the 1st cosine basis function.

The reason they look like that is because the IDCT reconstructs the signal by scaling and adding the cosine basis functions one by one. The constant component comes from the DC coefficient

F(0)

, and the sinusoidal component comes from the AC coefficient

F(1)

. The magnitudes and shapes of these components are determined by the corresponding DCT coefficients.

The Top Part of Figure 8 F(u)F(u)F(u) For a 1-D Signal f(i)f(i)f(i)

The Top Part of Figure 8 $F(u)$ For a 1-D Signal $f(i)$