Circuits and Systems, 2013, 4, 252-263
http://dx.doi.org/10.4236/cs.2013.43034 Published Online July 2013 (http://www.scirp.org/journal/cs)
High Speed Digital Oscillator Implementations Based on
Advanced Arithmetic and Architecture Techniques
Ali Shatnawi, Mufleh Shatnawi
Department of Computer Engineering, Jordan University of Science and Technology, Irbid, Jordan
Received September 24, 2012; revised December 1, 2012; accepted December 9, 2012
Copyright © 2013 Ali Shatnawi, Mufleh Shatnawi. This is an open access article distributed under the Creative Commons Attribu-
tion License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
ABSTRACT
The advances of digital arithmetic techniques permit computer designers to implement high speed application specific
chips. The currently produced digital circuits have demonstrated high performance in terms of several criteria, such as,
high clock rate, short input/output delay, small silicon area, and low power dissipation. In this paper, we implement
several sinusoidal generation methods to optimize their performance and output using advanced digital arithmetic tech-
niques. In this paper, the implementations of advanced digital oscillator structures with and without pipelining are pro-
posed. The synthesis results of the implementation with pipelining have proven that it is superior to other sinusoidal
generation methods in terms of the maximum frequency and signal resolution. Hence, this method is used in the design
of the proposed digital oscillator chip.
Keywords: Digital Oscillator; Single-Output Digital Oscillator; Multiple-Output Digital Oscillator;
Advanced Arithmetic Techniques; Hardware Implementation
1. Introduction
A sinusoidal oscillator can be designed using analog or
digital components. The sinusoidal parameters such as
the frequency, amplitude and phase, are easier to control
in digital oscillators than in analog ones. On the other
hand, the amount of harmonic distortion in digital oscil-
lators is higher than that in analog ones. This is obviously
attributed to the resulting quantization error and digital
arithmetic round-offs.
Digital sinusoidal oscillators are used in many applica-
tions, including communications, music synthesis, con-
trol, radar, and several digital signal processing applica-
tions. As the design and fabrication of digital integrated
circuits are getting very systematic and relatively simple,
the choice of the digital approach for sinusoidal oscilla-
tors has become very desirable.
The development of high speed digital signal process-
ing units is very important in the implementation of
real-time applications. This paper uses advanced digital
arithmetic techniques along with advanced hardware
design techniques to produce high-speed, high-frequency
and digital oscillator chips. The organization of the paper
will be as follows. Section 2 presents the motivation for
this work, Section 3 discusses some of the approaches
used in the design of digital oscillators, Section 4 intro-
duces the number representation used, Section 5 presents
some digital oscillators found in the literature, Section 6
analyses the oscillators used, Section 7 gives the imple-
mentation and results, and Section 8 concludes the paper.
2. Motivation
In the fields of communications, digital signal processing,
and digital image processing, which are known to be
computationally intensive, the performance of an imple-
mentation is decided by the efficiency of the critical
arithmetic operations. Like most DSP applications, the
performance of a digital oscillator is highly dependent on
the efficiency of the multiplication unit. In other words,
the optimization of the multiplication algorithm is critical
in the implementation of the digital sinusoidal oscillators;
especially, when the design is based on recursive algo-
rithms.
In this paper, we will work on two issues to implement
an efficient digital oscillator. In the first, we will device a
methodology to implement a very fast multiplier, a criti-
cal component of the digital oscillator. In the second, we
propose pipelined implementation of the entire design to
enable a fast clock rate.
C
opyright © 2013 SciRes. CS

A. SHATNAWI, M. SHATNAWI 253
3. Digital Oscillator Generation Methods
In literature, several methods to implement a digital os-
cillator had been proposed. Among the most known
methods are the following.
3.1. Look-Up-Table (LUT)
The samples of a general sinusoidal signal are stored in a
ROM-like memory, and re-generated at a rate that is
necessary to produce the desired frequency [1,2].
3.2. Recursive Method
In this method, a second-order recursive digital filter is
used to generate the sequence of the output samples of
the required waveform [3-10]. One of these methods is
based on the following difference equation.
 
2cos12 ,0ynynyn n  

(1)
where yis the output sample number n, and

n

2cos

is a multiplier coefficient. The value of this
coefficient plays an important role in determining the
frequency of the generated sinusoidal signal. The Z-
transform of (1) is given by:
 


 
1
12
1 2y
zz





2cos
12cos
zy
Yz


 (2)
Let the initial condition

10y

, then




12
2
s
y
zz






12
co
Yz (3)
But

1
12
sin
zz







sin 12
cos
z
Zn
 
 (4)
By comparing (3) with (4), the Z-inverse of (3) will be
given by
  
2sin 1 yn
sin
yn




2π
SS
TfT
(5)
Equation (5) shows that θ carries information about
the frequency of the sinusoidal signal generated by the
digital oscillator. The angle θ can be rewritten as:

  (6)
where TS is the time interval between consecutive sam-
ples of the generated sinusoidal signal, and f the desired
frequency. The value of TS determines the maximum
frequency of the digital oscillator chip, that is


s
f

1
s
T. The generated sinusoidal signal is represented in a
discrete form with a number of samples per cycle that is
given by
2π
N (7)

Combining (6) and (7), the number of samples can
also be expressed as
1
N
s
f
T
 (8)

The above equations show that the frequency of the
generated sinusoidal signal has a linear relationship with
the maximum frequency of the digital oscillator.
4. Fixed-Point Number Representation and
Fraction Manipulation
Some of the commercially available digital signals proc-
essing units have no hardware support for floating-point
arithmetic, usually called Floating Point Unit (FPU), due
to the cost limitation. Instead, they use software emula-
tion for the implementation of floating-point operations.
This can significantly limit the rate at which iterative
real-time applications can execute.
The cost of digital operations and the simplicity of the
arithmetic design are highly affected by the choice of the
numbering representation system. The fixed-point num-
ber representation is typically used when the hardware
cost, speed, and hardware complexity are considered in
the implementation of DSP processing units [3].
It should be noted that any digital arithmetic operation
using fixed-point representation can be performed as
integer digital arithmetic operation. This significantly
improves the execution speed of the digital signal proc-
essing units and alleviates the software complexity due to
software emulation used for implementing floating-point
operations. On the other hand fixed-pint arithmetic suf-
fers from the quantization error due to the finite-preci-
sion representation. However, in most digital signal
processing application, such as digital oscillators and
digital filters, the data and coefficients can be scaled to
reduce the effect of the quantization error on the accu-
racy of the implemented systems [11].
In this study, we have chosen the sign-magnitude fixed
point number representation for the implementation of
the digital oscillator. It has been shown that the hardware
implementation of sign-magnitude fixed point number
representation is required less area compare to 1’s com-
plement or 2’s complement [11]. This can be clearly seen
that only one inverter is required to generate a negative
number.
The format of the number representation, which is
shown in Figure 1, is composed of a sign bit, I bits for
the integer part, and F bits for the fraction part.
The number of fraction bits determines the upper and
lower bounds of both the frequency range and the total

A. SHATNAWI, M. SHATNAWI 254
S I F
S: sign
I: number of integer bits
F: number of fraction bits
Figure 1. The sign-magnitude fixed-point number repre-
sentation.
harmonic distortion of the implemented oscillator as will
be shown later.
5. Digital Arithmetic Techniques and
Related Work
The multiplication is a basic arithmetic operation in al-
most all digital signal processing applications. Digital
signal processing systems require hardware multipliers to
implement DSP algorithms efficiently. The speed of the
multiplier directly impacts the speed of the digital proc-
essing units [12-14].
There are many fast multipliers in the literature that
can be used in the design of an efficient digital oscillator
[11,15-18]. Regardless of the multiplication method, the
basic operation of a multiplication algorithm is the addi-
tion. In this section we consider the most common adder
structures. To further optimize the addition operations of
a multiplication, multi-operand addition techniques are
used [19].
5.1. Carry Save Adder (3:2 Adder/3:2 Counter)
The Carry Save Adder (CS A) is classified as a redundant
number system adder. The CSA Adder is used to add
more than two numbers—binary vectors-together, say x,
y, and z, and generate two binary numbers—vectors-
such that CS 
x
yz SC

. This addition can be
done in O(1) time, since the carry bits are saved in the C
vector and no carry propagation is required. The result of
the multi-operand addition can be obtained by adding the
finial binary vectors

SC
together using a conven-
tional number system adder like the carry look-ahead
adder.
The full adder (FA) can be thought of as a 1-bit CSA,
as shown in Figure 2 [14,19].
5.2. 4:2 Compressor
The “4-2 carry-save module” was proposed in [20]. It
contains a combination of FA cells, which are intercon-
nected to generate the output signal faster than the tradi-
tional CSA Adder, 3:2 Adder. The structure actually
compresses five bits (input signals) into three bits (output
signal); however, it is connected in such a way that four
of the inputs are coming from the same bit position and
one from the preceding bit position (known as carry-in,
Cin). The output of such a 4:2 Compressor consists of one
bit in the succeeding bit position (known as carry-out,
Cout) and two output bits in the same bit position. The
basic 4:2 Compressor structure is depicted in Figure 3.
This Compressor is widely used in a multiplier unit,
since it compresses four partial product bits into two bits,
and produces one carry bit. The structure of the com-
pressor reduces the number of partial product bits by one
half at each stage. The speed of such a 4:2 Compressor is
represented by the speed of three XOR gates in series
[15,21].
5.3. Multiplier Design
The digital arithmetic multiplication techniques can be
divided into three stages: partial products generation
stage, partial products summation stage, and the final
addition stage. In this section we explain different im-
plementations for the second stage. The second stage is
the most critical in the determination of the overall speed
of the multiplier. In this stage we can use one of
multi-operand addition techniques (Linear tree, Wallace
tree, and Compressor tree) to design a multiplier with the
minimum area and a low latency. The final stage; how-
ever, is an addition using one of the fastest conventional
adders, such as the carry look-ahead adder [11].
5.4. Array of Full Adder (FA)
In the multiplier, an array of FA’s is used to add the par-
tial products. The multiplier (X) and multiplicand (A) are
Figure 2. Full adder to carry save adder.
Figure 3. 4:2 Compressor.

A. SHATNAWI, M. SHATNAWI 255
unsigned integers with the same word-length, n bits. The
product requires 2n bits to be represented, as shown in (9)
(We refer to such multiplication by n-by-n multiplica-
tion).

11
00
22
2
nn
ij
ij
ij
ij
ij
A
a
Pxa






11
00
nn
ij
PX
Px






(9)
5.5. Linear Carry-Save Adder (CSA) Tree
The multiplier generates partial products using the modi-
fied Booth's recoding algorithm proposed in [14]. The
generated partial products are summed up using Linear
Carry-Save Adder Tree (LCSAT) as shown in Figure 4.
In each level, we generate two bits of the final product
result. The Carry Look-ahead Adder (CLA) is used in the
final stage to generate the remaining bits of the result
[11].
5.6. Wallace CSA Tree
In the multiplication operation, the addition of the partial
products is the most time consuming process. We present
a multi-operand addition technique to handle the proce-
dure of repetitive addition, which is referred to as the
Wallace tree, illustrated in Figure 5.
A Wallace Tree is composed of (3, 2) counters [14]. It
achieves the highest degree of parallelism with a delay
that grows with

logOn
32 . Traditionally, Wallace
Trees were not embraced by designers, because they are
much harder to design and layout due to their irregular
structures [11,14].
5.7. Compressor Tree
The 4:2 Compressor Tree has a more regular structure
than the ordinary CSA tree. It is made of “3:2 Counter”
counters, as the partial products are added up in the form
of a binary tree. The compressor structure can be ex-
tended to the wanted number of partial products by cre-
ating a new structure or combining some existing ones.
In [16-18], the proposed algorithms differentiate be-
tween the fast and the slow inputs and outputs of the 3:2
CSA counter. Moreover, the 4:2 Compressor adders were
used in the partial product summation tree, so as to re-
duce the delay of the parallel multipliers.
6. Digital Oscillator Analysis and Related
Work
The most common methods to generate sinusoidal sig-
nals are the recursive methods using a second-order dif-
ference Equation (1). It has been shown in [1-10] that
Figure 4. Multiplier based on the Linear Carry-Save adder.
Figure 5. Adding 9-operands using WCSA.
using recursive method, it is difficult to generate a given
desired frequency, and this difficulty increases as the
desired frequency is reduced where the sensitivity and
round-off errors increase by reducing the frequency.
Many of recursive digital oscillators have been proposed
in literature to increase the range of frequency that can be
generated and to reduce the effect of round-off errors.
These digital oscillators can be classified into the fol-
lowing:
 Single-Output Direct-Form Digital Oscillator
 Multiple-Output Direct-Form Digital Oscillator
 Complex Digital Oscillator with Integrator
 Combined Digital Oscillator
The direct-form oscillator is a single output oscillator
that generates a sinusoidal signal using a second-order
difference Equation (1) suffers from a round-off quanti-

A. SHATNAWI, M. SHATNAWI 256
zation error. The use of recursive methods increases the
impact of this error on the output accuracy in the worst
case, the quantization errors may accumulate to unde-
sired levels, leading to major discrepancy between the
generated output and the ideal output.
To improve the output signal generated from the di-
rect-form oscillator is a single output oscillator, the feed-
back circuit of the first or second order is used to reduce
the effect of the round-off error and increase the number
of samples of the generated sinusoidal signal. The feed-
back circuit has improved the generated sinusoidal signal
by reducing the amount of noise in the generated signal
[3].
The Multiple-Output Direct-Form Digital Oscillator
generates sine and cosine waveforms, namely


yn

c
yn
s
and , with a stable amplitudeand fixed phase
shift π2
e
. The multiple-output direct-form digital oscil-
lator suffers from a round-off quantization error; the
round-off error can be reduced by using a second order
error feedback as shown in [4].
The combined digital oscillator depends on combining
both oscillators that are capable of generating both the
real (cosine) and imaginary (sine) components of the
sinusoidal signal

j
. It should be emphasized that each
oscillator in the combined digital oscillator can generate
sinusoidal signals independently from the other oscillator.
In other words, the amplitude and frequency of each os-
cillator output can be designed separately.
The combined digital oscillator is capable to generate
sinusoidal signals with lower frequencies compared to
the multiple-output direct-form oscillator. This implies
that the generated signal from the combined digital os-
cillator has more samples than the generated signal from
individual multiple-output direct-form oscillator. It should
be emphasized that the multiple-output direct-form os-
cillators that are used to build the combined digital oscil-
lator have no error feedback circuits [5].
An enhancement on structure of combined digital os-
cillator has been introduced in [6] as shown in Figure 6.
The proposed digital oscillator represents the multiplier
coefficient


2cos

using b + 2 bits; 1 bit for the sign,
1 bit for the integer part and b bits for the fractional part.
It has been shown that the number of samples per cycle
depends on the smallest value of θ, which is given by

12 2
2
b







1
2
2
3, 4,, 2
b
b
m



1
min cos
  (10)
In the proposed structure, the value of the multiplier
coefficient can be increased in steps of 2−b over a specific
range as given by:
cos
1, 2,
m
m


 (11)
Figure 6. Block diagram of combined digital oscillator pro-
posed in [6].
where


m represents a quantized value of the actual
analog value of θ at a certain value of m. When the value
of m is small,


m will be close to π2. The combined
digital oscillator evaluates the difference between the
phases of both sinusoidal signals generated by the multi-
ple-output direct-form oscillator and the two sinusoidal
differential components,



sin π2n and

cos π2n.
The phase of the generated sinusoidal signal will then be
evaluated using the formula


π2 



g
m. The value
of

g
is very small in comparison with both and π2


m; therefore, the generated sinusoidal signal has a rela-
tively large number of samples, 2π


g
. N
e
The complex digital oscillator with integrator structure
consists of two digital integrators and two multipliers,
arranged in a closed-loop fashion proposed in [7]. The
gains for the integrators and multipliers coefficient val-
ues have been equivalently distributed over the whole
structure of the digital oscillator to increase the fre-
quency resolution and simplify the implementation of the
structure. The proposed structure has a quantization error
that is less than that of the direct-form digital oscillator.
The real and imaginary components (the cosine and sine)
of the signal

j
are generated.
The linear relationship between the generated fre-
quency of sinusoidal signal and the maximum operation
frequency of digital oscillator chip has been proven in [10],
the combined digital oscillator of the multiple-output
direct-form oscillator is proposed. The generated fre-
quency values are expressed by:


2
2
2π
1, 2,,21
b
dclk
b
m
ff
m

 (12)







where clk
f
represents the maximum operating frequency.
It is shown in (12) that the generated frequency and the
maximum frequency are linearly related.
7. Hardware Implementation and Results
The digital oscillator structures discussed in the previous
section is described in VHSIC hardware description lan-
guage (VHDL) [22,23]. To validate the functionality of

A. SHATNAWI, M. SHATNAWI 257
these oscillators, we have simulated their implementa-
tions using the Mentor Graphic simulation tool Model
Sim [24]. After simulation, we synthesized the structures
using the Xilinx synthesis tool. The Synthesis process
takes the conceptual VHSIC Hardware Description Lan-
guage (VHDL) design definitions, and generates the
logical or physical representation for the targeted silicon
device. The FPGA implementation using VIRTEX-5
family was chosen for the synthesis process [25].
7.1. The Use of Pipelining
Pipelining is an advanced hardware technique that, in
most cases, significantly improves the performance of
digital arithmetic units. The cost of pipelining is usually
embedded in the inter-stage buffers (registers). The per-
formance gain is attributed to the large throughput ob-
tained by breaking up the computation path over multiple
logic stages with shorter latencies. This enables a faster
clock rate but at the expense of a higher hardware com-
plexity [26,27].
7.2. Synthesis Parameters
The following synthesis metrics are considered.
 Maximum combinational path delay: This metric
represents the total combination delay of the critical
path of the digital arithmetic unit.
 Number of slice Look-Up-Tables (LUT): The look-
up-table represents the direct form implementation of
a Boolean function. This metric measures the number
of slices used for implementing the look-up-table. It
is usually used as an indicator to the utilized layout
area.
 Number of slice registers: This metric measure the
number of slices used for implementing the required
registers, which also indicates the utilized area.
 Number of fully used bit slices: This metric meas-
ures the actual number of used slices.
 Maximum frequency: The maximum operating fre-
quency of the digital arithmetic unit.
It should be emphasized that the FPGAs are built from
a grid of cells, called Configurable Logic Block (CLBs).
Each CLB contains a number of Look-Up Tables (LUTs)
and Flip-Flops. This grid is arranged into slices.
7.3. The Sign-Magnitude Multiplier
Implementation and Synthesis Results
The multiplication algorithms have been studied in de-
tails for four fast multiplier schemes in terms of the
maximum combinational path delay and the number of
slice LUTs. Tables 1 and 2 summarize the results for
different multiplier schemes.
The synthesis results (Figure 7) show that the Com-
Table 1. Sign-magnitude multipliers, maximum combina-
tional path delay (ns).
Number of bits
9 bits (s, 8) 17 bits
(s, 16) 33 bits (s, 32)
Array of full
adder 13.157 25.487 49.986
LCSA 8.810 13.680 23.397
Wallace tree 10.411 14.743 21.376
Multiplier schema
Compressor
tree 8.975 13.415 21.233
Table 2. Sign-magnitude mu ltipliers, numb er of slices LUTs.
Number of bits
9 bits
(s, 8) 17 bits (s, 16) 33 bits (s, 32)
Array of full
adder 99 401 1573
LCSA 118 421 1618
Wallace tree 124 446 1561
Multiplier schema
Compressor
tree 130 422 1647
0
5
10
15
20
25
30
35
40
45
50
55
9 bits (s,8)17 bits (s,16)33 bits (s,32)
Maximum Combinational Path Delay (ns)
Array of full adderLCSA Wallace treeCompressor tree
Figure 7. Multiplier results: maximum combination path
delay (ns) vs word sizes for different multiplier schemes.
pressor tree and the Wallace tree multipliers are superior
to the other multipliers in terms of the maximum combi-
national path delay for large number of bits. It is also
shown that the Compressor tree multiplier slightly out-
performs the Wallace Tree multiplier.
Figure 8 shows that increasing the number of bits re-
sults in an increase in the number of slice LUTs. This
increase is due to the need for a larger input/output
look-up table when the number of bits is increased. Ap-
proximately all multiplier schemes require almost the
same number of slice LUTs.
The Compressor tree multiplier has been used in im-
plementing all the digital oscillator structures. It outper-

A. SHATNAWI, M. SHATNAWI 258
0
200
400
600
800
1000
1200
1400
1600
1800
9 bits (s,8)17 bits (s,16)
Number of Slice LUTs
33 bits (s,32)
Array of full adderLCSA Wallace treeCompressor tree
Figure 8. Multiplier results: number of slice LUTs vs multi-
plier schemes for different word size.
forms all other multiplier schemes in terms of the maxi-
mum combinational path delay without a significant in-
crease in the number of slice LUTs. The next section
presents some digital oscillator implementation using the
Compressor tree multiplier.
7.4. Digital Oscillator Implementation and
Synthesis Results
The implementation of the four recursive digital oscilla-
tors mentioned earlier is presented in this section. Some
digital oscillator structures are implemented with a sec-
ond-order error feedback as proposed in [3]. The error
feedback is introduced to alleviate the quantization error
and increase the number of samples per cycle. This will
be further explained in the simulation section.
To distinguish between the digital oscillator structures,
we give name them as follows.
 Single-output oscillator: It produces a single output.
If feedback is employed, it will be called “single-
output oscillator with feedback”, and if pipelining is
employed it will be preceded by the word “pipelined”.
The main structure of this oscillator was proposed in
[3].
 Multiple-output oscillators: They are based on the
digital oscillator structures proposed in [5]. This os-
cillator can also be constructed with error feedback or
without error feedback and with or without pipelin-
ing.
 Basic combined oscillator: It is based on the oscillator
proposed in [6]. This oscillator can be constructed
using two symmetric multiple-output oscillators.
 Basic complex oscillator and advanced complex os-
cillator: They are based on the oscillators proposed in
[7,9], respectively. Similarly, if pipelining is used
these digital oscillators will be called “basic complex
pipelined oscillator” and advance complex pipelined
oscillator, respectively.
 Advanced combined oscillator: It is it constructed us-
ing a single modified multiple-output oscillator struc-
ture, arranged with a finite-state machine [10]. This
oscillator can have the same variations as in the basic
digital oscillator.
The pipeline structures for all the digital oscillators
mentioned earlier are proposed in this paper. The digital
oscillator structures with pipelining are compared with
the digital oscillator structures without pipelining for all
digital oscillator combinations. The digital oscillator
structures are implemented with different word-size: 8,
16, and 32 bits.
Figure 9 shows the value of maximum frequency for
different digital oscillator structure types with different
word sizes. From these results, we can deduce the fol-
lowing.
 As number of bits is increased, the maximum fre-
quency of each of the digital oscillator structures is
decreased.
 The pipelined oscillator structures have higher maxi-
mum frequency than the non-pipelined structures.
 Considering the pipelined versions of all digital os-
cillator structures (except the basic and advanced
digital oscillator), the maximum frequency of the
digital oscillator without error feedback is slightly
higher than that of the digital oscillator with error
feedback.
Figures 10 and 11 show the number of slice register
and the number of fully used bit slices for different os-
cillator structures, and for different number of bits, re-
spectively. Figures 10 and 11 imply the following.
 Both the number of fully used bit slices and the num-
ber of slice registers show the same behavior for the
different oscillator structures.
 As the number of bits is increased, the number of
fully used bit slices of all digital oscillator structure
types increases. This is also true for the number of
slice registers.
 All pipelined oscillator structures have higher num-
bers of fully used bit slices than the non-pipelined os-
cillator structures. This is also valid for the number of
slice registers. This is a natural consequence of the
use of pipelining, as it requires extra inter-stage reg-
isters.
Figure 12 shows the number of slice LUTs versus the
different oscillator structure and for different number of
bits. We can deduce the following.
 As number of bits is increased, the number of slice
LUTs of all digital oscillator structures is increased.
This is also valid for the number of slice registers.
 The pipelining does not affect the number of slice
LUTs in all oscillator structure for all word sizes.
7.5. Digital Oscillator Simulation Results
This sinusoidal signal is generated using the following
assumptions.

A. SHATNAWI, M. SHATNAWI 259
0
25
50
75
100
125
150
175
200
225
250
275
Si ng l e -
Output
Oscillator
Without
Feedback
Multiple-
Output
Os cillator
Without
Feedback
Basic
Combi ned
oscillator
Without
Feedback
Advance
Comb ined
oscillator
Without
Feedback
Sing le -
Outpu t
Os cillator
With
Feedback
Multiple-
Outp ut
Oscillator
With
Feedback
Basic
Combi ned
oscillator
With
Feedback
Advance
Combined
Oscillat or
With
Feedback
Bas
Com
Maxi mum Frequency(MHz)
i c
plex
Advance
Complex
Without Pipelining (8)Without Pipelining (16)Without Pipelining (32)
With Pipelining (8)With Pipelining (16)With Pipelining (32)
Figure 9. Maximum frequency vs oscillator types for different word sizes.
0
100
200
300
400
500
600
700
800
900
1000
1100
Sing le-
Output
Os cilla to r
Without
Feedback
Multiple-
Output
Oscillator
Without
Feedback
Basic
Comb ine d
oscillator
Without
Feedback
Advance
Combined
oscillator
With out
Feedback
Sing l e-
Outp u t
Oscillator
With
Feedback
Multiple-
Output
Oscillator
With
Feedback
Basic
Combi ned
oscillator
With
Feedback
Advance
Comb ine d
Oscillator
With
Feedback
Ba
Com
Number of Slice Register
sic
plex
Advance
Complex
Without Pipelining (8)Without Pipelining (16)Without Pipelining (32)
With Pipelining (8)With Pipelining (16)With Pipelining (32)
Figure 10. Number of slice register vs oscillator types for different word sizes.
 The number format is (s, 1, 7), that is, seven bits for
the fraction part, one bit for the integer part, and one
bit for the sign.
 The multiplier coefficient 2cos

is determined us-
ing 22os255  . Hence, the theoretical value for
both ,
c
F

N are given by:



11
255 256cos0.996093755.066
360 5.06671.062
t
t
N

 

cos 
(13)
 The initial condition


2yis assumed to be equal to
(−11).
Figure 13 shows the sinusoidal signal generated by
the single-output direct-form oscillator without the 2nd
level error feedback. The generated number of samples in
a complete cycle is 2π


g
g
N, where

g
is the gen-
erated phase. 46

g and
N7.826


71
g. The generated
number of samples is less than the expected theoretical
number of samples due to the quantization error.
Figure 14 shows the sinusoidal signal generated by
the single-output direct-form digital oscillator with 2nd
level error feedback. The sinusoidal signal has a number
of samples that is approximately equal to theoretical
umber of samples. n

g
N5.07 and


g.

A. SHATNAWI, M. SHATNAWI 260
0
100
200
300
400
500
600
Singl e -
Output
Oscillator
Without
Fee dback
Multiple-
Outp u t
Oscillator
Without
Feedback
Basic
Com bined
oscillator
Without
Feedback
Advance
Comb ined
oscillator
Without
Feedbac k
Singl e -
Output
Oscilla tor
With
Feedback
Multiple-
Outp u t
Oscillator
With
Feedback
Basi c
Combined
oscill a tor
With
Feedback
Advance
Combined
Os cillato r
With
Feedback
Ba
Com
Number of fully used Bit Slices
sic
plex
Advance
Complex
Without Pipelining (8)Without Pipelining (16)Without Pipelining (32)
With Pipelining (8)With Pipelining (16)With Pipelining (32)
Figure 11. Number of fully used bit slices vs oscillator types for different word sizes.
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
Singl e -
Output
Oscillator
Without
Feedback
Multiple-
Output
Os cillator
Without
Feedback
Basic
Comb ine d
oscill a t or
Without
Feedback
Advance
Combined
oscillator
With out
Feedback
Singl e-
Outp ut
Oscillator
With
Feedbac k
Multiple-
Outp ut
Oscillator
With
Feedback
Basic
Com bined
oscillator
Wit h
Feedback
Advance
Comb ine d
Oscilla tor
With
Feedback
Ba
Com
Number of S lice LUTs
sic
plex
Advance
Complex
Without Pipelining (8)Without Pipelining (16)Without Pipelining (32)
With Pipelining (8)With Pipelining (16)With Pipelining (32)
Figure 12. Number of slice LUTs vs oscillator types for different word sizes.
The sinusoidal signal generated by the advanced com-
bined digital oscillator is shown in Figure 15. The multi-
ple-output direct-form digital oscillator is used to gener-
ate a sinusoidal signal with the following parameters:
 Multiplier coefficient 2cos

: It is computed using
cos 7  . The theoretical value for both , 22
F

N,
are then given by:



1
cos7 256
4375 88.433
4.071




1
cos0.0273
360 88.433
t
t
N



(14)
 Initial condition


2y: It is given by


2

y
248.
The advance combined digital oscillator depends on
the phase difference between π2 and the generated
phase from several multiple-output direct-form digital
oscillators. The above combined digital oscillator gener-
ates a sinusoidal signal with


88.4331.567
360 1.567230
π2
N
  

(15)
8. Conclusions
We have utilized advanced architecture and arithmetic
techniques to implement digital oscillator.
The synthesis results have shown that the pipelined

A. SHATNAWI, M. SHATNAWI 261
Y(n)
-70
-50
-30
-10
10
30
50
70
1611 16 21 26 31 36
Amplitude
41 46
Y(n)
Figure 13. Sinusoidal signal generated by a single-output direct-form digital oscillator without 2nd level error feedback.
-150
-100
-50
0
50
100
150
161116 21 26 31 36 41 46 5156
Ampl i t ude
61 66 71
Y(n)
Figure 14. Sinusoidal signal generated by a single-output direct-form digital oscillator with 2nd level error feedback.
-300
-200
-100
0
100
200
300
0
49 98
147 196 245
Amplitude
294
Ys YcYd
Figure 15. Sinusoidal signal generated by an advanced combined digital oscillator.

A. SHATNAWI, M. SHATNAWI 262
igital oscillator str
llator structures h
sh
e arith-
m
[1] M. Schanerbehe Implementation
ductures are superior in terms of the
maximum frequency when compared with the non-pipe-
lined ones. This has led to a significant enhancement of
the generated sinusoidal signal in terms of the frequency
and number of samples per cycle.
The simulation results of osciave
own that the combined digital oscillators proposed in
[10] have produced sinusoidal signals with a large num-
ber of samples in comparison with the other digital os-
cillators. This makes the combined digital oscillator
structure proposed in [10] the preferable digital oscillator
structure among all digital oscillator structures.
It is to be noted that our work is sensitive to th
etic algorithms used. Thus, if faster arithmetic algo-
rithms are proposed, new implementations for digital
oscillators should be devised.
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