How many bits does it take to store a 3-minute song using an audio encoding method without compression that samples at the rate of 40,000 samples per second and has a bit depth of 16? what if it uses a compression scheme with a compression ratio of 5: 1?

a) 51,200 bits.  b) : 10,240 bits.

Explanation:

If we sample a 3 - minute song  at 40,000 samples per second, and we are told that each sample is represented by a combination of 16 bits, we can calculate the number of bits needed to store the song as follows:

N = 40,000 samples /sec . 16 bits /sample. 180 sec,. = 51,200 bits.

Now, if we compress the samples taken, after quantization, in a 5:1 relationship, this means that we will store only 1 of each 5 samples, essentially removing redundant information, so the number of bits needed after compresssion will be 51,200 / 5 = 10,240 bits.

It is interesting to note that if we apply the Sampling Theorem, that says that in order to be recoverable, a signal must be sampled at least two times during the shortest cycle, or at a frequency double than the highest frequency component in the signal, we can see that the signal is sampled at 20 Khz, which is usually the upper audible frequency limit.