Suppose we have 32 samples from a mono audio file.
(a) If the audio was sampled at $8 \mathrm{kHz}$, how many seconds does this represent?
(b) If we carry out a DCT with blocksize equal to 32, how many DCT coefficients do we end up with?
(c) Numbering the DCT coefficients from 0,
i) What frequency, in $\mathrm{Hz}$, does coefficient number 1 represent?
ii) What frequency, in $\mathrm{Hz}$, does coefficient number 16 represent?

Question

Quizplus · Accepted Answer

The Answer is (a)

32/8000 \mathrm{sec}==4 \mathrm{msec}

(b) 32
(c) i)

1/2

cycle per 32 samples

=1/2

cycle per

32 / 8000

sec

=1/2 /(32/8000)

cycles-per-sec

=4000 / 32

cycles-per-sec

=125 \mathrm{~Hz}

$(a) 32/8000 \mathrm{sec}==4 \mathrm{msec} (b) 32 (c) i) 1/2 cycle per 32 samples =1/2 cycle per 32 / 8000 sec =1/2 /(32/8000) cycles-per-sec =4000 / 32 cycles-per-sec =125 \mathrm{~Hz} Fig. 8.1: DCT for a 1-D signal; reconstruction using only F(0) as opposed to only F(0) and F(1) . ii) =8 cycles per 32/8000 \mathrm{sec} =8 /(32/8000) cycles-per-sec =2 \mathrm{kHz} (Note that the max frequency we analyze for is 4 \mathrm{kHz} .) BEGIN not\_on\_exam: \begin{aligned} & \# ext { S-plus language script. } \ & \# ext { 1Ddct.s } \ & ext { blocksize }=32 \ & ext { pi }=3.14159265 \ & ext { factor }=2.0 / ext { blocksize } \ & ext { denom }=\left(2.0^{*} ext { blocksize } ight) / p i \ & ext { Lambda }= ext { function }(u)\{ \ & ext { if }(u==0) ext { return }(1 / \operatorname{sqrt}(2))\ & ext{else return(1)}\ & \}~\# ext{end of Lambda} \end{aligned} END not on exam.$ Fig. 8.1: DCT for a 1-D signal; reconstruction using only

F(0)

as opposed to only

F(0)

and

F(1)

.
ii)

=8

cycles per

32/8000 \mathrm{sec}

=8 /(32/8000)

cycles-per-sec

=2 \mathrm{kHz}

(Note that the max frequency we analyze for is

4 \mathrm{kHz}

.)
BEGIN not\_on\_exam:

\begin{aligned}& \# ext { S-plus language script. } \& \# ext { 1Ddct.s } \& ext { blocksize }=32 \& ext { pi }=3.14159265 \& ext { factor }=2.0 / ext { blocksize } \& ext { denom }=\left(2.0^{*} ext { blocksize } ight) / p i \& ext { Lambda }= ext { function }(u)\{ \& ext { if }(u==0) ext { return }(1 / \operatorname{sqrt}(2))\& ext{else return(1)}\& \}~\# ext{end of Lambda}\end{aligned}

$(a) 32/8000 \mathrm{sec}==4 \mathrm{msec} (b) 32 (c) i) 1/2 cycle per 32 samples =1/2 cycle per 32 / 8000 sec =1/2 /(32/8000) cycles-per-sec =4000 / 32 cycles-per-sec =125 \mathrm{~Hz} Fig. 8.1: DCT for a 1-D signal; reconstruction using only F(0) as opposed to only F(0) and F(1) . ii) =8 cycles per 32/8000 \mathrm{sec} =8 /(32/8000) cycles-per-sec =2 \mathrm{kHz} (Note that the max frequency we analyze for is 4 \mathrm{kHz} .) BEGIN not\_on\_exam: \begin{aligned} & \# ext { S-plus language script. } \ & \# ext { 1Ddct.s } \ & ext { blocksize }=32 \ & ext { pi }=3.14159265 \ & ext { factor }=2.0 / ext { blocksize } \ & ext { denom }=\left(2.0^{*} ext { blocksize } ight) / p i \ & ext { Lambda }= ext { function }(u)\{ \ & ext { if }(u==0) ext { return }(1 / \operatorname{sqrt}(2))\ & ext{else return(1)}\ & \}~\# ext{end of Lambda} \end{aligned} END not on exam.$ END not on exam.

Suppose We Have 32 Samples from a Mono Audio File 8kHz8 \mathrm{kHz}8kHz

Suppose We Have 32 Samples from a Mono Audio File $8 \mathrm{kHz}$