From cdc388c67094e1bd686407637150dcb9edeff628 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Christian=20M=C3=BCnker?= Date: Tue, 12 Dec 2023 09:17:05 +0100 Subject: [PATCH] =?UTF-8?q?Update=20formatting=20Signed-off-by:=20Christia?= =?UTF-8?q?n=20M=C3=BCnker=20?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- docs/source/manual/fixpoint_arithmetics.rst | 54 +++++++++++---------- docs/source/manual/index.rst | 8 +-- 2 files changed, 33 insertions(+), 29 deletions(-) diff --git a/docs/source/manual/fixpoint_arithmetics.rst b/docs/source/manual/fixpoint_arithmetics.rst index 8c0093bb5..69b370675 100644 --- a/docs/source/manual/fixpoint_arithmetics.rst +++ b/docs/source/manual/fixpoint_arithmetics.rst @@ -7,10 +7,12 @@ Fixpoint Arithmetics Overview --------- -In contrast to floating point numbers, **fixpoint** numbers have a fixed scaling, +In contrast to floating point numbers, **fixpoint** numbers have a fixed scaling, requiring more care to avoid over- or underflows. The same binary word can -represent an integer or a fractional number, signed or unsigned. The binary point -and whether the MSB represents the sign bit or not, is in the designers head ... +represent an integer (:numref:`fig_twos_complement_int`) or a fractional +(:numref:`fig_twos_complement_frac`) number, signed or unsigned. The position of the +binary point and whether the MSB represents the sign bit or not, it is all in the +designer's head ... .. _fig_twos_complement_int: @@ -46,7 +48,7 @@ limit cycles in recursive filters. Typical simulation results are shown in :numref:`fig_pyfda_screenshot_yn_overflows`, where first the input signal exceeds the numeric range and then the output signal. -The overflow behaviour is set to 'wrap', resulting in twos-complement wrap around +The overflow behaviour is set to 'wrap', resulting in twos-complement wrap around with changes in the sign. .. _fig_pyfda_screenshot_yn_overflows: @@ -55,23 +57,41 @@ with changes in the sign. :alt: Screenshot of fixpoint simulation results (time domain) :width: 80% :align: center - + Fixpoint filter response with overflows +Sign extension +*************** + +When increasing the number of integer bits, numbers need to be sign extended, +i.e. the new leading bits need to be filled with the sign bit +(:numref:`fig_sign_extension`). Extending the +number of fractional bits just requires zero padding. + +.. _fig_sign_extension: + +.. figure:: ../img/manual/sign_extension.png + :alt: Sign extension of integer and fractional numbers + :width: 40% + :align: center + + Sign extension of integer and fractional numbers + Overflow behaviour ******************* -When adding two numbers or when reducing the number of integer bits, the number may +After summation or when reducing the number of integer bits, the result may not fit in the numeric range. Discarding one or more leading bits to obtain the desired wordlength is easy but may produce wrap-arounds. The resulting sign changes can introduce instability and limit-cycle -oscillations to the system. +oscillations to the system (:numref:`fig_fixpoint_overflow`, left-hand side). -Saturation is much more benign but requires a little more effort: When summing two numbers, +Saturation (:numref:`fig_fixpoint_overflow`, right-hand side) is much more benign +but requires a little more effort: Before adding two numbers, both need to be sign extended by one bit to enable overflow detection. As shown in :numref:`fig_fixpoint_overflow`, when the two leading bits (sign and carry) are `01` -or `10`, the result exceeds the numeric range and needs to be replaced by the maximum +or `10`, the result exceeds the numeric range and needs to be replaced by the maximum resp. minimum representable value. When reducing the number of integer bits, similar checks need to be performed to test for overflows. @@ -85,22 +105,6 @@ checks need to be performed to test for overflows. Overflow behaviour with wrap-around or saturation -Sign extension -*************** - -When increasing the number of integer bits, numbers need to be sign extended, -i.e. the new leading bits need to be filled with the sign bit. Extending the -number of fractional bits just requires zero padding. - -.. _fig_sign_extension_int: - -.. figure:: ../img/manual/sign_extension.png - :alt: Sign extension of integer and fractional numbers - :width: 40% - :align: center - - Sign extension of integer and fractional numbers - Truncation and rounding ************************** diff --git a/docs/source/manual/index.rst b/docs/source/manual/index.rst index 96fd9d65c..6249503e1 100644 --- a/docs/source/manual/index.rst +++ b/docs/source/manual/index.rst @@ -18,7 +18,7 @@ specification and parameters for the filter / system to be designed resp. analyz .. toctree:: :maxdepth: 1 - :caption: Subwindow for Input Widgets + :caption: **Subwindow for Input Widgets** input_specs input_coeffs @@ -26,11 +26,11 @@ specification and parameters for the filter / system to be designed resp. analyz input_info input_fixpoint_specs - The subwidgets on the right-hand side allow for graphical analyses of the system. +The subwidgets on the right-hand side allow for graphical analyses of the system. .. toctree:: :maxdepth: 1 - :caption: Subwindow for Plotting Widgets + :caption: **Subwindow for Plotting Widgets** plot_hf plot_phi @@ -43,7 +43,7 @@ Some documentation treats general filter design and fixpoint arithmetics stuff. .. toctree:: :maxdepth: 1 - :caption: General Documentation + :caption: **General Documentation** fixpoint_arithmetics