Thesis Proposal for: Companding in Fixed Point DSPs

1
Thesis Proposal forCompanding in Fixed Point
DSPs

• A scheme to reduce the quantization (roundoff)
  errors in Fixed Point DSPs
• Ari Klein

2
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

3
CONSTANT ENVELOPE
LARGE SIGNALS

• 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

4
If there is only an ADC and a DAC (no DSP in
between)
u(nT)
y(nT)
g(n)
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

5
What about with a DSP between the ADC and
DAC? Might consider
y(nT)
u(nT)
g(n)
1/g(n)
Would this scheme work?
6
NO! this will horribly distort the input/output
behavior
y(nT)
u(nT)
g(n)
1/g(n)

• Taking a DSP, multiplying its input by a
  time-varying signal g(n), and then multiplying
  the output by 1/g(n) will, in general, CHANGE the
  input/output behavior of the DSP (dramatically)

7
For example, let y(n)u(n-k) So the DSP is a
k-sample delay. If I do
y(nT)
u(nT)
g(n)
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

8
Must make internal corrections in DSP to ensure
that there is no change in input/output
behavior. Consider a system given
by x(n1)Ax(n)Bu(n) y(n)Cx(n)Du(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. Tsividiss Equations w(n1) G(n1)A
G-1(n)w(n)G(n1)Bu(n) y(n)CG-1(n)w(n)
Du(n) This is a new internally time-varying
system, whose states are always large, but whose
input/output behavior is identical to the
original systems!
9
Companding for DSPs (in principle)
input
output
G(n) generator
10
Single-g Companding G(n) g(n) I
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

11
Single-g Companding G(n) g(n) I
Starting from Professor Tsividiss Equations
And setting G(n) g(n) I gives
w(n1) d(n) (Aw(n)Bg(n)u(n)
) g(n)y(n) Cw(n)Dg(n)u(n)
g(n1)
where d(n)
g(n)

• Companding system has
• Input g(n)u(n)
• Output g(n)y(n)

12
Single-g Companding G(n) g(n) I
w(n1) d(n) (Aw(n)Bg(n)u(n)
) g(n)y(n) Cw(n)Dg(n)u(n)
In general,
Can be implemented as
Z-1
Envelope Detection
Z-1
13
(No Transcript)
14
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
15
Multi-g Companding G(n) is diagonal, but with
unequal elements on diagonal
DSP with companding
G(n) generator

• More complicated than single-g companding
• But works for any system

The G(n) generator could be (for example)
DSP without companding
Envelope Detection
16
Simplifications for multi-g companding
Consider a k-delay element
Z-k
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,
17
Thesis Proposal

• Implement companding on some actual DSPs
• I have already implemented it on the simple
  digital reverberator shown below

18
Thesis ProposalExplore Possible Improvements
For example
Taking advantage of masking by using a bank of
bandpass filters
19
Thesis ProposalTheoretical Considerations
Already Proven Companding does not change the
original systems

• 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

20
Thesis ProposalImplementing 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

21
Thesis ProposalQualitative 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,
  non-companding DSP versus the companding DSP?
• Conduct objective listening tests, similar to
  psychoacoustic experiments

22
Thesis Proposal Quantitative Comparisons

• Quantify the improvements as a function of signal
  level and number of bits

• Mean square error of output
• SNR
• Harmonic distortion in spectrum

For example
If these are NOT good metrics for quantifying
improvements due to companding, I will also need
to FIND some good metrics