NCO
Generate real or complex sinusoidal signals
 Library:
DSP HDL Toolbox / Signal Operations
DSP HDL Toolbox / Sources
Description
The NCO block generates real or complex sinusoidal signals, while providing hardwarefriendly control signals.
A numericallycontrolled oscillator (NCO) accumulates a phase increment and uses the quantized output of the accumulator as the index to a lookup table that contains the sine wave values. The wrap around of the fixedpoint accumulator and quantizer data types provide periodicity of the sine wave, and quantization reduces the necessary size of the table for a given frequency resolution.
For an example of how to generate a sine wave using the NCO block, see Generate Sine Wave. For more information on configuration and implementation, refer to the Algorithms section.
The block also provides these features:
Optional framebased output.
A lookup table compression option to reduce the lookup table size. This compression results in less than one LSB loss in precision. See Lookup Table Compression for more information.
An optional input port for external dither.
An optional reset port that resets the phase accumulator to its initial value.
An optional output port for the current NCO phase.
Ports
Note
This block appears in the Sources libraries with Phase increment source parameter set to
Property
. The only input port is valid.This block appears in the Signal Operations libraries with Phase increment source parameter set to
Input port
. This configuration shows the optional input port inc.
This icon shows the optional ports of the NCO block.
Input
inc
— Phase increment
scalar integer
Phase increment, specified as a scalar integer. The block casts this value to match the accumulator word length.
double
and single
data types are
supported for simulation but not for HDL code generation.
Dependencies
To enable this port, set the Phase increment
source parameter to Input
port
.
Data Types: single
 double
 int8
 int16
 int32
 int64
 uint8
 uint16
 uint32
 uint64
 fixdt([],N,0)
offset
— Phase offset
scalar integer
Phase offset, specified as a scalar integer.
double
and single
data types are
supported for simulation but not for HDL code generation.
Dependencies
To enable this port, set the Phase offset
source parameter to Input
port
.
Data Types: single
 double
 int8
 int16
 int32
 int64
 uint8
 uint16
 uint32
 uint64
 fixdt([],N,0)
dither
— Dither
integer  column vector of integers
Dither, specified as an integer or a column vector of integers. The length of the vector must equal the Samples per frame parameter value.
double
and single
data types are
supported for simulation but not for HDL code generation.
Dependencies
To enable this port, set the Dither source
parameter to Input port
.
Data Types: single
 double
 int8
 int16
 int32
 int64
 uint8
 uint16
 uint32
 uint64
 fixdt([],N,0)
valid
— Control signal that enables NCO operation
scalar
Control signal that enables NCO operation, specified as a Boolean
scalar. When this signal is 1
, the block increments
the phase and captures any input values. When this signal is
0
, the block holds the phase accumulator and
ignores any input values.
When the Samples per frame parameter is greater
than 1
, this value enables processing of
Samples per frame samples.
Data Types: Boolean
reset accum
— Resets the accumulator
scalar
Control signal that resets the accumulator, specified as a Boolean
scalar. When this signal is 1
, the block resets the
accumulator to its initial value. This signal does not reset the output
samples in the pipeline.
Dependencies
To enable this port, select the Enable accumulator reset input port parameter.
Data Types: Boolean
Output
sin, cos, exp
— Generated waveform
scalar  column vector
Generated waveform, returned as a scalar or as a column vector with
length equal to the Samples per frame parameter
value. The output can be a single port that returns
sin or cos values, a
single port that returns exp values representing
cosine + j*sine
, or two ports that return
sin and cos values,
respectively.
When all input values are fixedpoint type or all input ports are disabled, the block determines the output type using the Output data type parameter. When any input value is floatingpoint type, the block ignores the Output data type parameter. In this case, the block returns the waveform as floatingpoint values. Floatingpoint data types are supported for simulation but not for HDL code generation.
Dependencies
By default, this output port is a sine wave, sin. The port label and format changes based on the Type of output signal parameter.
phase
— Current phase of NCO
scalar  column vector
Current phase of NCO, returned as a scalar or as a column vector with
length equal to the Samples per frame parameter
value. The phase is the output of the quantized accumulator with offset
and increment applied. If quantization is disabled, this port returns
the output of the accumulator with offset and increment applied. The
values are of type fixdt(1,N,0)
, where
N
is the Number of quantizer
accumulator bits parameter value. If quantization is
disabled, then N
is the Accumulator Word
length parameter value.
If any input value is floatingpoint type, the block returns the phase as a floatingpoint value. Floatingpoint data types are supported for simulation but not for HDL code generation.
Dependencies
To enable this port, select the Enable phase port parameter.
Data Types: single
 double
 fixdt(1,N,0)
valid
— Indicates validity of output data
scalar
Control signal that indicates validity of output data, returned as a
Boolean scalar. When output valid is
1
, the values on the sin,
cos, exp, and
phase ports are valid. When output
valid is 0
, the values on
the output ports are not valid.
When the Samples per frame parameter is greater
than 1
, this signal indicates the validity of all
elements in the output vector.
Data Types: Boolean
Parameters
Main
Note
This block supports double
and single
input for simulation but not for HDL code generation. When all input values are
fixedpoint type or all input ports are disabled, the block determines the
output type using the Output data type parameter. When any
input value is floatingpoint type, the block ignores the Output data
type parameter. In this case, the block returns the waveform and
optional phase as floatingpoint values.
To use the FixedPoint Designer™ data type override feature, you can obtain a
double
output value by applying double
input data to one of the optional ports.
Phase increment source
— Source of phase increment
Input port
(default)  Property
You can set the phase increment with an input port or by entering a
value for the parameter. If you select
Property
, the Phase
increment parameter appears for you to enter a value. If
you select Input port
, the
inc port appears on the block.
Phase increment
— Phase increment for generated waveform
100
(default)  integer
Phase increment for the generated the waveform, specified as an integer. The block casts this value to match the accumulator word length.
Dependencies
To enable this parameter, set the Phase increment
source parameter to
Property
.
Data Types: single
 double
 int8
 int16
 int32
 uint8
 uint16
 uint32
 fixdt([],N,0)
Phase offset source
— Source of phase offset
Input port
(default)  Property
You can set the phase offset with an input port or by entering a value
for the parameter. If you select Property
,
the Phase offset parameter appears for you to enter
a value. If you select Input port
, the
offset port appears on the block.
Phase offset
— Phase offset for generated waveform
0
(default)  integer
Phase offset for the generated waveform, specified as an integer.
Dependencies
To enable this parameter, set the Phase offset
source parameter to
Property
.
Data Types: single
 double
 int8
 int16
 int32
 uint8
 uint16
 uint32
 fixdt([],N,0)
Dither source
— Source of number of dither bits
Property
(default)  Input port
 None
You can set the dither from an input port or from a parameter. If you
select Property
, the Number of
dither bits parameter appears. If you select
Input port
, a port appears on the block.
If you select None
, the block does not add
dither.
Number of dither bits
— Bits used to express dither
4
(default)  positive integer
Number of dither bits, specified as a positive integer.
Dependencies
To enable this parameter, set the Dither
source parameter to
Property
.
Samples per frame
— Vector size for framebased input and output
1
(default)  positive integer
When you set this value to 1
, the block has scalar
input and output. When this value is greater than 1
,
the dither port expects a column vector of length
Samples per frame and the
sin, cos,
exp, and phase ports
return column vectors of length Samples per
frame.
Enable look up table compression method
— Compress the lookup table
off
(default)  on
By default, the block implements a noncompressed lookup table, and the output of this block matches the output of the NCO block. When you enable this option, the block implements a compressed lookup table. The Sunderland compression method reduces the size of the lookup table, losing less than one LSB of precision. The spurious free dynamic range (SFDR) is empirically 1–3 dB lower than the noncompressed case. The hardware savings of the compressed lookup table allow room to improve performance by increasing the word length of the accumulator and the number of quantize bits. For detail of the compression method, see Algorithms.
Enable accumulator reset input port
— Enable reset control signal
off
(default)  on
Select this parameter to enable the reset accum
port. When reset accum is 1
, the
block resets the accumulator to its initial value.
Type of output signal
— Format of output waveform
Sine
(default)  Cosine
 Complex exponential
 Sine and cosine
If you select Sine
or
Cosine
, the block shows the applicable
port, sin or cos. If you
select Complex exponential
, the output is of
the form cosine + j*sine
and the port is labeled
exp. If you select Sine and
cosine
, the block shows two ports,
sin and cos.
When you set the Type of output signal parameter
to Complex exponential
or Sine
and cosine
, the block implements a 1/8 sine wave
lookup table for each of the sine and cosine parts of the waveform, and
uses control logic to select and invert the values to generate both sine
and cosine waveforms. This optimization means that dual output mode uses
similar hardware resources compared to single output mode.
Enable phase port
— Output current phase
off
(default)  on
Select this parameter to return the current NCO phase on the phase port. The phase is the output of the quantized accumulator, with offset and increment applied. If quantization is disabled, this port returns the output of the accumulator, with offset and increment applied.
Data Types
Rounding Mode
— Rounding mode for fixedpoint operations
Floor
(default)
Rounding
mode for fixedpoint operations. Rounding
Mode is a readonly parameter with value
Floor
.
Overflow mode
— Overflow mode for fixedpoint operations
Wrap
(default)
Overflow mode for fixedpoint operations. Overflow mode is a readonly parameter. Fixedpoint numbers wrap around on overflow.
Accumulator Data Type
— Accumulator data type
Binary point
scaling
(default)
Accumulator data type description. This parameter is readonly, with
value Binary point scaling
. The block defines
the fixedpoint data type using the Accumulator
Signed, Accumulator Word length, and
Accumulator Fraction length parameters.
Accumulator Signed
— Signed or unsigned accumulator data format
Signed
(default)
This parameter is readonly. All output is signed format.
Accumulator Word length
— Accumulator word length
16
(default)  integer
Units are in bits. This value must include the sign bit.
If you clear the Quantize phase parameter, then Accumulator word length determines the LUT size. For HDL code generation, the LUT size must be between 2 and 2^{17} entries. When you select Enable look up table compression method, this parameter must be an integer in the range [5,21]. When you clear Enable look up table compression method, this parameter must be an integer in the range [3,19]. For more information on how this parameter affects the LUT size, see the Algorithms section.
When you select the Quantize phase parameter, there is no limit to the Accumulator word length parameter value.
Accumulator Fraction length
— Accumulator fraction length
0
(default)  integer
This parameter is readonly. The accumulator fraction length is zero bits.
The accumulator operates on integers. If the phase increment is
fixdt
type with a fractional part, the block
returns an error.
Quantize phase
— Quantize accumulated phase
off
(default)  on
When you select Quantize phase, the block quantizes the result of the phase accumulator to a fixed bitwidth. The block uses this quantized value to select a waveform value from the lookup table. Quantizing the output of the phase accumulator enables you to reduce the lookup table size without lowering the frequency resolution. Select the size of the lookup table by using the Number of quantizer accumulator bits parameter.
When you clear Quantize phase, the block uses the full accumulator value as the address of the lookup table.
Number of quantizer accumulator bits
— Number of quantizer accumulator bits
12
(default)  integer
Number of quantizer accumulator bits, specified as an integer scalar less than the accumulator word length. For HDL code generation, this parameter value must result in a LUT size between 2 and 2^{17} entries. When you select Enable look up table compression method, this parameter must be an integer in the range [5,21]. When you clear Enable look up table compression method, this parameter must be an integer in the range [3,19]. For more information on how this parameter affects the LUT size, see the Algorithms section.
Dependencies
To enable this parameter, select the Quantize phase parameter.
Output Data Type
— Output data type
Binary point
scaling
(default)  double
 single
Specify the data type for the sin, cos, and exp ports. This parameter is ignored if any input is of floatingpoint type. In that case, the output data type is floatingpoint.
If you select Binary point scaling
, the
block defines the fixedpoint data type using the Output
Signed, Output Word length, and
Output Fraction length parameters.
Output Signed
— Signed or unsigned output data format
Signed
(default)
This parameter is readonly. All output is signed format.
Output Word length
— Output word length
16
(default)  integer
Units are in bits. This value must include the sign bit.
Output Fraction length
— Output fraction length
14
(default)  integer
Units are in bits.
Model Examples
Algorithms
The frequency resolution of the sine wave depends on the size of the accumulator. Given a sample time, T_{s}, and the desired output frequency resolution Δf, calculate the necessary accumulator word length, N.
$$N=\mathrm{ceil}\left({\mathrm{log}}_{2}\left(\frac{1}{{T}_{s}\cdot \Delta f}\right)\right)$$
For a desired output frequency F_{o}, calculate the phase increment.
$$phaseincrement=\mathrm{round}({F}_{0}{T}_{s}{2}^{N})$$
Quantizing the output of the phase accumulator enables you to reduce the lookup table size without lowering the frequency resolution. Calculate the quantized word length to achieve a desired spurious free dynamic range (SFDR).
$$Q=\mathrm{ceil}\left(\frac{SFDR12}{6}\right)$$
Phase offset and dither are optionally added at the accumulator stage. For a desired phase offset (in radians) of the output waveform, calculate the phase offset value that the block adds in the accumulator.
$$phaseoffset=\frac{{2}^{N}\cdot desiredphaseoffset}{2\pi}$$
The NCO implementation depends on whether you select Enable look up table compression method.
Without lookup table compression, the block uses the same quartersine lookup table as the NCO block. The size of the LUT is 2^{Q2}×W bits, where Q is Number of quantizer accumulator bits and W is Output word length.
The block casts the phase increment value to match the accumulator word length.
If you do not enable Quantize phase, then Q = N, where N is Accumulator Word length. Consider the impact on simulator memory and hardware resources when you select these parameters.
When you set the Type of output signal parameter to
Complex exponential
or Sine and
cosine
, the block implements a 1/8 sine wave lookup table for each of
the sine and cosine parts of the waveform, and uses control logic to select and invert
the values to generate both sine and cosine waveforms. This optimization means that dual
output mode uses similar hardware resources compared to single output mode.
For an example of how to generate a sine wave using the NCO block, see Generate Sine Wave.
Lookup Table Compression
When you select lookup table (LUT) compression, the NCO block applies the Sunderland compression method. Sunderland techniques use trigonometric identities to divide each phase of the quarter sine wave into three components and express it as:
$$\mathrm{sin}(A+B+C)=\mathrm{sin}(A+B)\mathrm{cos}(C)+\mathrm{cos}(A)\mathrm{cos}(B)\mathrm{sin}(C)\mathrm{sin}(A)\mathrm{sin}(B)\mathrm{sin}(C)$$
If the quartersine phase has Q2
bits, then
the phase components A and B have a word
length of
LA=LB=ceil((Q2)/3)
.
Phase component C contains the remaining phase bits. If the phase
has 12 bits, then the quarter sine phase has 10 bits, and the components are defined
as:
A, the four most significant bits
$$(0\le A\le \frac{\pi}{2})$$
B, the next four bits
$$(0\le B\le \frac{\pi}{2}\times {2}^{4})$$
C, the remaining two least significant bits
$$(0\le C\le \frac{\pi}{2}\times {2}^{}{}^{8})$$
Given the relative sizes of A, B, and C, the equation can be approximated by:
$$\mathrm{sin}(A+B+C)\approx \mathrm{sin}(A+B)+\mathrm{cos}A\mathrm{sin}C$$
The NCO block implements this equation with one LUT for $$\mathrm{sin}(A+B)$$ and one LUT for $$\mathrm{cos}(A)\mathrm{sin}(C)$$. The second term is a fine correction factor that you can truncate to fewer bits without losing precision. Therefore, the second LUT returns a fourbit result.
With the default accumulator size of 16 bits, and the default quantized phase width of 12 bits, the LUTs use 2^{8}×16 plus 2^{6}×4 bits (4.5 kb). For comparison, a quartersine lookup table without compression uses 2^{10}×16 bits (16 kb). The compression approximation is accurate within one LSB, resulting in an SNR of at least 60 dB on the output. See [1].
When you set the Type of output parameter to
Complex exponential
or Sine and
cosine
, the block implements a compressed lookup table for each of
the sine and cosine parts of the waveform. The hardware resource use is still
smaller than dual output mode with an uncompressed table.
Control Signals
The block has two input control signals, reset accum
(optional) and valid, and one output control signal,
valid. When reset accum is
1
, the block sets the phase accumulator to its initial value.
When the input valid is 1
, the block
increments the phase and captures any input values. When this signal is
0
, the block holds the phase accumulator and ignores any
input values. When the output valid signal is
1
, the values on the other output ports are valid.
Latency
The latency of the NCO block is six cycles.
Performance
This table shows postsynthesis resource utilization for the HDL code generated for the NCO block in the Generate Sine Wave example. The synthesis targets a Xilinx^{®} ZC706 (XC7Z045ffg9002) FPGA.
Resource  Uses 

LUT  744 
Slice Reg  156 
Xilinx LogiCORE DSP48  0 
After place and route, the maximum clock frequency of the design is 477 MHz.
References
[1] Cordesses, L., "Direct Digital Synthesis: A Tool for Periodic Wave Generation (Part 1)." IEEE Signal Processing Magazine. Volume 21, Issue 4, July 2004, pp. 50–54.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.
This block supports C/C++ code generation for Simulink^{®} accelerator and rapid accelerator modes and for DPI component generation.
HDL Code Generation
Generate Verilog and VHDL code for FPGA and ASIC designs using HDL Coder™.
HDL Coder™ provides additional configuration options that affect HDL implementation and synthesized logic.
This block has one default HDL architecture.
ConstrainedOutputPipeline  Number of registers to place at
the outputs by moving existing delays within your design. Distributed
pipelining does not redistribute these registers. The default is

InputPipeline  Number of input pipeline stages
to insert in the generated code. Distributed pipelining and constrained
output pipelining can move these registers. The default is

OutputPipeline  Number of output pipeline stages
to insert in the generated code. Distributed pipelining and constrained
output pipelining can move these registers. The default is

When you set Dither source to
Property
, the block adds random dither every cycle. If you generate a validation model with these settings, a warning is displayed. Random generation of the internal dither can cause mismatches between the models. You can increase the error margin for the validation comparison to account for the difference. You can also disable dither or set Dither source toInput port
to avoid this issue.You cannot use the NCO block inside a Resettable Synchronous Subsystem (HDL Coder).
Version History
Introduced in R2013aR2022a: Moved to DSP HDL Toolbox from DSP System Toolbox
Before R2022a, this block was named NCO HDL Optimized and was included in the DSP System Toolbox™ DSP System Toolbox HDL Support library.
R2022a: Resource optimization for dual output mode
When you set the Type of output signal parameter
to Complex exponential
or Sine and
cosine
, this block implements a 1/8 sine wave lookup table for
each of the sine and cosine parts of the waveform, and uses control logic to select
and invert the values to generate both sine and cosine waveforms. This optimization
means that dual output mode uses hardware resources similar to single output mode.
In previous releases, the block implemented one lookup table for each output
waveform.
R2020a: Valid input port required
In previous releases, the input validIn port was optional. It is now required and renamed valid. If you are using no other input ports, the block uses the valid signal as an enable signal.
R2020a: Phase quantization for floatingpoint input
The output waveform returned from floatingpoint input values has changed. The output waveform now matches the waveform returned from the same input values specified in fixedpoint types.
Before R2020a, when using floatingpoint input types, the block did not quantize the phase internally. The block expected floatingpoint phase increment and phase offset inputs specified in radians. Now, the block quantizes the phase internally, and you must specify the input phase increment and offset in terms of the quantized size, for both floatingpoint and fixedpoint input types.
For example, before R2020a, for a floatingpoint HDL NCO to generate output samples with a desired output frequency of F_{0} and sample frequency of F_{s}, you had to specify the phase increment as 2π(F_{0}/F_{s}) and phase offset as π/2.
Starting in R2020a, you must specify the phase increment and phase offset in terms of the quantized size, N. These input values are the same as the input values you use with fixedpoint types. Specify the phase increment as (F_{0}×2^{N})/F_{s} and the phase offset as (π/2)×2^{N}/2π, or 2^{N}/4.
R2020a: LUTRegisterResetType option removed
In previous releases, you could choose from two options for the
LUTRegisterResetType parameter on the HDL Block
Properties dialog box. The two options were
default
and none
. Starting
in R2020a, the block ignores the parameter setting and uses
none
for this parameter value. This option does not
connect a reset signal to the LUT registers. This configuration enables the
synthesis tool to determine whether to implement the lookup tables with LUTs or
BRAM.
R2020a: Highthroughput interface
You can generate framebased waveforms from the block. The block returns a vector where each element represents a sample in time. Set the Samples per frame parameter to the desired output vector size.
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