Thesis Proposal for:
Companding in Fixed Point DSPs
A scheme to reduce the quantization
(roundoff) errors in Fixed Point DSPs
Ari Klein
ADVANTAGE OF LARGE
SIGNALS
• When the signals in the DSP are large, they
take full advantage of the available bits.
• When the signals in the DSP are small, only
a few of the available bits are used.
• The roundoff error is essentially
independent of the signal level, so the
distortion due to roundoff is much worse for
small signals
LARGE SIGNALS
CONSTANT ENVELOPE
•To reduce roundoff error, all signals should be “as
large as possible” whenever digital
•Since the system has a set overflow tolerance, “as
large as possible” means that all digital signals
should have a roughly constant envelope, where the
constant is slightly lower than the system overflow
tolerance
If there is only an ADC and a DAC (no DSP in between):
u(nT)
g(n)
X
ADC
X
DAC
y(nT)
1/g(n)
•Since the input is NOT generally constant envelope, multiply
the input by a time-varying signal, g(n), BEFORE the ADC.
g(n) should be LARGE when the input signal is SMALL, and
SMALL when the input signal is LARGE
•To make the output equal to the input, the output must be
multiplied by 1/g(n)
•The multiplication by 1/g(n) must be done AFTER the DAC, so
that the signal is large whenever it is digital
What about with a DSP between the ADC and DAC?
Might consider:
u(nT)
g(n)
X
ADC
DSP
DAC
1/g(n)
Would this scheme work?
X
y(nT)
NO! this will horribly distort the
input/output behavior
u(nT)
g(n)
X
ADC
DSP
DAC
X
y(nT)
1/g(n)
Taking a DSP, multiplying its input by a timevarying signal g(n), and then multiplying the
output by 1/g(n) will, in general, CHANGE the
input/output behavior of the DSP (dramatically)
For example, let y(n)=u(n-k)
So the DSP is a k-sample delay.
If I do:
u(nT)
g(n)
X
ADC
Z-k
DAC
X
y(nT)
1/g(n)
•The output is now g(n-k)*u(n-k)/g(n)
•This is NOT equal to u(n-k).
•It is DISTORTED by the factor g(n-k)/g(n).
•This is a time-varying distortion.
•More complicated systems will have more complicated distortions
Must make internal corrections in DSP to ensure that
there is no change in input/output behavior.
Consider a system given by:
x(n+1)=A*x(n)+B*u(n)
y(n)=C*x(n)+D*u(n)
I want the internal states to be large for all n,
so set w(n)=G(n)*x(n)
Doing the algebra gives Prof. Tsividis’s Equations:
w(n+1)= [G(n+1)*A* G-1(n)]*w(n)+[G(n+1)*B]*u(n)
y(n)=[C*G-1(n)]*w(n)+D*u(n)
This is a new internally time-varying system, whose states are
always large, but whose input/output behavior is identical to the
original system’s!
Companding for DSPs (in principle)
input
X
ADC
DSP
DAC
DAC
ADC
G(n)
generator
DAC
X
output
Single-g Companding: G(n) = g(n) * I
/
ADC
DSP
DAC
X
ADC
Envelope
Detection
•simple
•works well for systems where all the signals are “roughly” in
phase with each other
•set g(n) = k / env(n) where env(n) is the envelope of the
input, and k is a constant dependent on the overflow tolerance
Single-g Companding: G(n) = g(n) * I
Starting from Professor Tsividis’s Equations:
And setting G(n) = g(n) * I gives:
w(n+1) = d(n) * (A*w(n)+B*[g(n)*u(n)] )
[g(n)*y(n)] = C*w(n)+D*[g(n)*u(n)]
where d(n) =
g(n+1)
g(n)
Companding system has:
•Input = g(n)*u(n)
•Output = g(n)*y(n)
Single-g Companding: G(n) = g(n) * I
w(n+1) = d(n) * (A*w(n)+B*[g(n)*u(n)] )
In general,
[g(n)*y(n)] = C*w(n)+D*[g(n)*u(n)]
Can be implemented as:
Z-1
/
ADC
DSP
ADC
Envelope
Detection
Z-1
/
DAC
X
Unfortunately, single-g companding does NOT work well for
systems where some signals are large at the same time as other
signals are small
For such systems, need multi-g companding:
G(n) still diagonal, but with unequal elements on diagonal
Multi-g Companding
G(n) is diagonal, but with unequal elements on diagonal
X
ADC
DSP with
companding
DAC
X
DAC
ADC
G(n)
generator
The G(n) generator could be (for example):
DSP without
companding
Envelope
Detection
DAC
•More
complicated
than single-g
companding
•But works for
any system
Simplifications for multi-g companding
Consider a k-delay element:
If the input is large: gi(n) * xi(n)
the output is also large: gi(n-k) * xi(n-k)
Can get desired
Can just delay gi(n) by k,
Z-k
Thesis Proposal
•Implement companding on some actual DSPs
• I have already implemented it on the simple digital
reverberator shown below
Thesis Proposal
Explore Possible Improvements
For example:
Taking advantage of masking by using a bank of bandpass filters
Thesis Proposal
Theoretical Considerations
Already Proven:
Companding does not change the original system’s
•Stability
•Controllability
•Observability
Would like to find:
•General method for easily implementing multi-g companding
•The “optimal” G(n) for minimizing quantization errors
•Upper bound on performance improvement due to companding
•General description of the type of systems where companding is most useful
Thesis Proposal
Implementing in Hardware
Implement the companding system on an actual fixed-point
DSP in real-time
•Build the necessary analog components:
•Timeshare ADCs and DACs (current implementation requires 2
ADCs and 3 DACs – probably too expensive)
•Make them variable gain to “absorb” the multiplies and divides
•Get the Simulink model for the digital companding system to work
on the DSP in real time
•Prototype (non-companding) system
•Real-time digital envelope detection to create g(n) signals
•Companding system
Thesis Proposal
Qualitative Comparisons
• For a given output “quality” how many bits are
needed in the original, non-companding DSP
versus in the companding DSP?
• For a given number of bits, how does the output
“quality” compare between the original, noncompanding DSP versus the companding DSP?
• Conduct objective listening tests, similar to
psychoacoustic experiments
Thesis Proposal
Quantitative Comparisons
Quantify the improvements as a function of
signal level and number of bits
For example:
•Mean square error of output
•SNR
•Harmonic distortion in spectrum
If these are NOT good metrics for quantifying improvements due
to companding, I will also need to FIND some good metrics