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Hi, and welcome to our special
module on the MP3 audio encoder.

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MP3 is a shorthand for MPEG layer 3, and

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MPEG is a shorthand for
the Motion Picture Expert Group.

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And what this all means is
that at one point in the 90s,

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a lot of experts got together and
agreed on a set of standards for

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video and audio compression and encoding.

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MP3 turned out to be the most
used audio digital format for

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digital audio storage,
streaming and playback.

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And today a portable music device,

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thanks to the MP3 encoding,
can store up to 30,000 songs, which

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really means you can carry your entire
music collection with you everywhere.

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So in this video we will look at the
technology behind the success of MP3, and

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we will describe in detail
how the MP3 encoder works.

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You will see how all the tools that
you have learned in our DSB class from

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Fourier transform to filtering,
from sampling into quantization,

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they all come together
in this application.

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So how does the encoding and
decoding process take place?

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Supposed you start with a discrete
time sound of signal x[n].

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This is processed by the encoder and
converted into a binary string.

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The decoder will take
that binary string and

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convert it back in to a sound signal y[n].

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The goal of this encoding and

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decoding chain is to reduce the memory
requirements to store the sound waveform.

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And the real achievement of MP3
is its ability to greatly reduce

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the amount of data needed
to encode a file, and

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this at a very reasonable trade-off with
respect to sound quality degradation.

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The data reduction is determined
by looking at the amount of memory

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that is necessary to store
the output of the encoder,

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by comparing this quantity to the amount
of memory that we should have used

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if we had wanted to store the original
signal in uncompressed format.

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And remember that an uncoded raw audio
file will require quite a bit of storage.

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For instance, if we sample at 48
kilohertz, which is the DVD standard,

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and we use 16 bits per sample,

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we will need 12 megabytes to store
a single minute of audio in stereo.

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On the other hand, a high-quality
MP3 will require just 1.5 megabytes,

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which represents almost an order
of magnitude in data reduction.

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To achieve this performance, the coding
has to be done in a very clever way, and

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one of the key ingredients in MP3 is
a model of the human auditory system.

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So MP3 does not attempt to
preserve the original framework.

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But rather it focuses on coding
the elements of the waveform

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that are most important to our way of
listening to music and hearing sounds.

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In particular,
the distortion introduced by the encoder,

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the loss of information introduced
by the encoding mechanism,

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is placed in parts of the spectrum of
the original signal that we cannot hear.

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We will see that in more
detail in just a minute.

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As we said, the origins of
MP3 date back to the 90s when

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the Moving Picture Expert Group,
in short MPEG,

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was set up by the International Standard
Organization to develop algorithms and

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standards for audio and video compression.

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And the audio compression part of
the standard, the MP3 protocol,

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had its origins in a set of
compression algorithms that had been

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developed in the 80s by
the Fraunhofer Institute in Germany.

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We see a photo of the team
here in this picture.

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The MP3 standard was quickly
embraced by the industry, and

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this widespread acceptance is what
decreed its success ultimately.

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Now let's try to understand how MP3
works using the simple block diagram.

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Your input signal x[n] enters
a bank of subband filters.

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There are 32 parallel filters
that subdivide the input signal

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into 32 independent channels that span
the full spectral range of the input.

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Each channel is then quantized
independently using a very clever method,

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and the quantized sample
are then formatted and

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encoded in a continuous bitstream.

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The quantization scheme is clever
because the number of bits

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allocated to each subband is dependent
on the perceptual importance

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of each subband with respect to
the overall quality of the audio waveform.

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In other words,

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subbands that are deemed by the
psycho-acoustic model not to be important,

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or are difficult to be perceived,
are allocated very few or no bits at all.

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Whereas the most perceptually relevant
subbands are allocated the bulk of

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the entire bit budget.

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The reason why we can safely
allocate different amounts

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of bits to the different subbands is to
be found in the so-called masking effect

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of the human auditory system.

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Suppose you have a sound with
a strong sinusoidal component,

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as in this picture here.

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The blue line represent
the spectrum of the sound, and

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here with the red dot we indicate
the strong sinusoidal component.

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When your ear listens to a sound
like this, a masking effect takes

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place whereby frequency components
in the vicinity of the dominant peak

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are not heard unless they are louder
than a given masking threshold.

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In this figure, for example,

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the masking threshold is
indicated by the red dotted line.

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And what it indicates is that anything in
the spectrum that falls below the red line

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will not be heard and

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therefore can be removed without
any loss of perceptual quality.

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Masking effects are something
that we experience every day.

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Imagine being in a perfectly quiet room,
like in your home at night.

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You can even hear your wristwatch ticking.

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But of course, you wouldn't be able to
hear that noise in normal conditions

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during the day, when a lot of other
auditory stimuli are reaching your ears.

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Although if you were to record the audio
environment and analyze its spectrum,

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you will see that it still contains the
information about your wristwatch ticking.

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The shape of the masking threshold
is a function of the loudness and

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the frequency of the dominant tone.

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And it has been determined experimentally

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by running a lot of listening
tests with human subjects.

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Masking in the human ear takes
place within the critical bands.

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And critical bands
are portions of the spectrum

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that are treated by
the ear as a single unit.

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Everything that happens within a critical
band cannot be further resolved

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by the ear, so two different frequencies
taking place in the same critical band

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are perceived as a single tone.

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There are approximately 24 critical
bands in the human ear, and

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here is a picture of their
distribution in frequency.

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As you can see,
they get wider as we go up in frequency,

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they follow a logarithmic scale.

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Which means that the resolution power of
the ear is stronger at low frequencies,

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whereas at high frequencies
we're less discriminant.

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And therefore, when we quantize
things across critical bands,

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we can probably fit more noise in the high
frequencies than in the low frequencies.

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In the end, the purpose of
the psycho-acoustic model is to compute

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the minimum number of bits that we need
to use to quantize each of the 32 subband

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filter outputs so that the perceptual
distortion is as little as possible.

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In the end we're given
a non-uniform bit allocation

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which will allocate fewer bits to
the bands where the masking is strongest.

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Interestingly enough, the specifications
of the psycho-acoustic model are not part

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of the MP3 standard, which means
that manufacturers of MP3 encoders

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can compete with better and better
versions of their psycho-acoustic model.

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In the end, the number of bits used for

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each subband is sent along with
the quantized data to the decoder, so

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it doesn't really matter how this
bit distribution has been generated.

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From a technical point of view, as you can
imagine, there are a lot of fine details

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in the inner workings of
the psycho-acoustic model and

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the bit allocation procedure.

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And we will not have time to examine
all of this in this presentation, but

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we can roughly sum up what happens inside
the psycho-acoustic model like so.

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First of all,

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remember that all processing is performed
on subsequent windows of a given length.

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So that input signal comes in and
the stream of input samples

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is cut into chunks of a given length,
say, 1,024 samples.

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An FFT is then used to estimate
the energy of the signal in each

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of the subbands computed
by the filter bank.

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For each subband we try to distinguish
between tonal and non-tonal components,

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components that have a strong sinusoidal
shape and noise-like components.

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We have looked at masking for
tonal components, but

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a similar type of masking takes place for

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non-tonal components, and we will have
to take that into account as well.

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The individual masking effect for tonal
and non-tonal components is computed for

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each critical band.

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And then these results are summed together

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to obtain a global masking curve for
the audio frame that we are analyzing.

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This masking curve is mapped
onto the 32 subbands.

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And the number of bits
that we will use for

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each subband is computed as a function
of the signal-to-mask ratio,

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the power of the signal versus
the masking power for each critical band.

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Let's now talk about the implementation
of this subband filtering in MP3.

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As we said, the input is split
across a filter bank that

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contains 32 filters isolating
different parts of the spectrum.

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These filters are implemented as
512-tap FIRs, and they're followed

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by a 32-times down-sampler to provide
the independent subband samples.

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The filter prototype is a simple
low pass with a cutoff frequency

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of pi over 64, and
a total bandwidth of pi over 32.

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The different subbands are obtained by
modulating the base filter with a cosine

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at multiples of pi over 64, and
the resulting filter bank looks like this.

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We're showing the positive
half of the frequency axis.

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This would be the first low
pass filter with an x zero.

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This is the second one,
the third, the fourth, and so

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on, covering the entire spectrum.

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Now let's go back to
the implementation of the filter bank.

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As you can see here from this block
diagram, each branch in the filter bank

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comprises an FIR filter of length 512 and
a 32-time downsampler here.

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What this means, of course, is that 31
out of 32 output samples of this filter

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are discarded, and so this is, of course,
a very wasteful implementation.

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Let's try and
make this a little bit more efficient.

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This is actually explained
in the MP3 standard.

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We start with the equation that expresses
the output of the subband number i as

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a convolution of the impulse response of
the filter for that branch with the input.

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And here you see that
the downsampling factor

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translates to a factor of 32
in front of the input index.

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We can now replace the expression for
the impulse response of the filter

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as the prototype impulse response
times the modulating factor that

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brings the filter to the proper
position in the frequency band.

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And then here we're going
to apply a little trick.

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We're going to express the index
k as the sum of two indices.

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Namely, we're going to say that k is

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equal to 64 times and index p plus q,

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where q ranges from 0 to 63 and
p ranges from 0 to 7.

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So with this split of the summation,
we can write the previous line as a double

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summation, for p that goes from 0 to 7 and
for q that goes from 0 to 63,

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as the same term, the modulation
term that we've seen before.

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The prototype impulse response and
the input.

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Where, again, we have made
the substitution k = 64 p + q.

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Okay, so

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with this trick we can actually simplify
the first term of this double summation.

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Consider the cosine term.

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We can write cosine of pi over 64 times 2i

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plus 1 times 64p plus some other term,
let's call it f,

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that depends only on i and q, and
we don't really care about that.

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Now here, 64 is canceled out and

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we're left with cosine of 2ip
pi plus p pi plus this term.

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And now, well, this is a multiple of 2 pi,
so it doesn't influence the angle.

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And here we have a multiple of pi.

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And we know that cosine of pi plus alpha
is equal to minus cosine of alpha.

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And so in the end,
what we can do is simplify this cosine

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as cosine of pi over 64,
times 2i plus 1 times q minus 16.

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And add a term minus 1 to the power
of p that we can move over to

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the second summation.

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And we have a simplified, quote unquote,
expression that looks like so.

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An outside sum here that only involves the
cosine modulation, and an inner sum here,

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which is a pre-subsampled implementation
of the filtering operation.

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If we work out the indices and

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convert this to an algorithmic procedure,
this is what we need to do.

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We will use a 512-tap
input circular buffer, and

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we will shift at each step 32 new input
audio samples, starting from the newest.

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So at any time,

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the circular buffer is holding 512
input samples in time-reversed order.

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Then we take a new 512-point buffer and
we fill it,

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sample by sample, with the product between
the prototype impulse response and

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the content of the circular buffer.

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Next we compute this
intermediate quantity here,

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which is the sum of the contents of
this new buffer, 64 points apart.

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We can do that for 63 different points.

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And if you do the math, there are seven
points that we have summed together for

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each q index.

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Finally, each subband output is given
by this sum here, where we're taking

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the intermediate quantity c[q] that
we computed before and we modulate it

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with the cosines at the frequencies
that we had defined in the beginning.

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And finally, quantization.

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This is where the great bit rate
savings are going to be achieved.

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MP3 uses uniform quantization
of subband samples.

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And the number of bits per
sample in each subband

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is determined by the psycho-acoustic
model, as we explained before.

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We also said before that MP3
works on subsequent audio frames,

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a frame being a window of input samples
that is processed independently.

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There are 36 samples per band and
per frame in the MP3 standard.

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And so

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since all of these 36 samples are going
to be quantized by the same quantizer,

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a rescaling is needed so that we are using
the full range of the quantizer.

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Remember how uniform quantization works.

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A quantizer maps an input interval
to a set of quantization levels.

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Of course, you have to make sure that
the range of your input signal matches

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the range of the quantizer.

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For instance, this quantizer expects
input to range from minus 1 to 1,

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but if your actual input only
lives in this small sub-interval,

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you will not be able to make use
of the full quantization range.

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So rescaling normally would imply
perfect renormalization of the 36

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samples by dividing the samples by
the largest sampling magnitude.

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Of course, in order for

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the decoder to then reconstruct
the actual levels of the input,

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we would have to send this normalization
factor alongside with quantized data.

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But this would require
a lot of side information.

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We would use 16 or 32 bits to encode
normalization factor and send it along.

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Instead, the MPEG standard defines
16 predefined scale factors.

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We will choose the one that best matches
the actual range of the input and

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only use 4 bits to communicate
this range to the decoder,

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thanks to the fact that these
predefined levels are set in stone.

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Finally, the actual quantization is
performed according to this formula,

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where b is the number of bits as
provided by the psycho-acoustic model,

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and Qa and Qb,
functions of the number of bits,

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are parameters that are encoded
inside the MP3 standard.

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Finally, let's listen to some examples.

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We all know that MP3 works very well, so

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what we want to concentrate on here
is the importance of the variable

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bit allocation across the subbands as
performed by the psycho-acoustic model.

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So for a fixed bit budget we
could choose to allocate the same

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number of bits to all subbands.

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This would be uniform bit allocation.

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Or we could use
a psycho-acoustic model and

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allocate the bits smartly across subbands.

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And so here are the examples,
starting with the original signal [NOISE].

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Now let's listen to the same
signal encoded with uniform bit

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allocation [NOISE].

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And finally, this is the result of
a full-fledged MP3 implementation with

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psycho-acoustically based
bit allocation [NOISE].

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Of course, in both the uniform and

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non-uniform bit allocation encoding
schemes, the target bit rate was very,

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very low in order to exacerbate
the effects of quantization.

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But the principle holds for all bit rates.