The document discusses active filters and provides information on different types of filters including:
- Butterworth filters which have a flat frequency response in the passband and stopband.
- Classification of filters such as low-pass, high-pass, and band-pass.
- Advantages of active filters over passive filters such as greater gain and flexibility.
- Design procedures for first and second order low-pass Butterworth filters including calculating cutoff frequencies from RC values.
Using Chebyshev filter design, there are two sub groups,
Type-I Chebyshev Filter
Type-II Chebyshev Filter
The major difference between butterworth and chebyshev filter is that the poles of butterworth filter lie on the circle while the poles of chebyshev filter lie on ellipse.
A complete description of including circuit diagram, gain equation, features of Instrumentational amplifier , its working principle, applications, practical circuits, Proteus simulation and conclusion.
Uet, Peshawar Pakistan
Batch-06
Using Chebyshev filter design, there are two sub groups,
Type-I Chebyshev Filter
Type-II Chebyshev Filter
The major difference between butterworth and chebyshev filter is that the poles of butterworth filter lie on the circle while the poles of chebyshev filter lie on ellipse.
A complete description of including circuit diagram, gain equation, features of Instrumentational amplifier , its working principle, applications, practical circuits, Proteus simulation and conclusion.
Uet, Peshawar Pakistan
Batch-06
Analysis of Butterworth and Chebyshev Filters for ECG Denoising Using WaveletsIOSR Journals
Abstract: A wide area of research has been done in the field of noise removal in Electrocardiogram signals.. Electrocardiograms (ECG) play an important role in diagnosis process and providing information regarding heart diseases. In this paper, we propose a new method for removing the baseline wander interferences, based on discrete wavelet transform and Butterworth/Chebyshev filtering. The ECG data is taken from non-invasive fetal electrocardiogram database, while noise signal is generated and added to the original signal using instructions in MATLAB environment. Our proposed method is a hybrid technique, which combines Daubechies wavelet decomposition and different thresholding techniques with Butterworth or Chebyshev filter. DWT has good ability to decompose the signal and wavelet thresholding is good in removing noise from decomposed signal. Filtering is done for improved denoising performence. Here quantitative study of result evaluation has been done between Butterworth and Chebyshev filters based on minimum mean squared error (MSE), higher values of signal to interference ratio and peak signal to noise ratio in MATLAB environment using wavelet and signal processing toolbox. The results proved that the denoised signal using Butterworth filter has a better balance between smoothness and accuracy than the Chebvshev filter. Keywords: Electrocardiogram, Discrete Wavelet transform, Baseline Wandering, Thresholding, Butterworth, Chebyshev
Implementation and comparison of Low pass filters in Frequency domainZara Tariq
Demonstrating the application results of some low pass filters in a frequency domain.
Pictures and MATLAB code been used in the experiment are taken from the internet.
In this presentation we discuss about the active filters and mentioned its frequency response along with block diagrams. Also discussed its pros and cons in this presentation.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
5. An analog filer has system fnction Ha(s)--a (a) (10 pts,) Comvert .pdfinfo324235
5. An analog filer has system fnction Ha(s)--a (a) (10 pts,) Comvert this analog filter into a
digital iker by means of the bilineasr filter by means of the bilinear trasformation method with T,
= 0.1. (b) (5 pts.) Is this filter FIR or IIR? (c) (5 pts.) Find the poles of this digital filher
Solution
Hundreds if not thousands of different kinds of filters have been developed to meet the needs of
various applications. Despite this variety, many filters can be described by a few common
characteristics. The first of these is the frequency range of their pass band. A filter\'s pass band is
the range of frequencies over which it will pass an incoming signal. Signal frequencies lying
outside the pass band are attenuated. Many filters fall into one of the following response
categories, based on the overall shape of their pass band.
Low-pass filters pass low-frequency signals while blocking high-frequency signals. The pass
band ranges from DC (0 Hz) to a corner frequency FC.
High-pass filters pass high-frequency signals while blocking DC and low-frequency signals. The
pass band ranges from a corner frequency (FC) to infinity.
Band-pass filters pass only signals between two given frequencies, blocking lower and higher
signals. The pass band lies between two frequencies, FL and FH. Signals between DC and FL are
blocked, as are signals from FH to infinity. The pass band of these filters is often characterized
as having a bandwidth that is symmetric around a center frequency.
Band-stop filters block signals occurring between two given frequencies, FL and FH. The pass
band is split into a low side (DC to FL) and a high side (FH to infinity). For this reason, it\'s
often simpler to specify a band-stop filter by the width and center frequency of its stop band.
Band-stop filters are also called notch filters, especially when the stop band is narrow.
Figure 1 shows how each of these filters operates on a swept-frequency input signal.
Figure 1. Filters are usually characterized by their frequency-domain performance. The effects
of a few common filter types on a swept-frequency input signal are shown here.
In the examples, the signal increases continuously in frequency, from a low frequency to a high
frequency. When the signal frequency is within the filter\'s pass band, the filter passes the signal.
As the signal moves out of the pass band, the filter begins to attenuate the signal.
Note that the transition from the pass band to the stop band is a gradual process, where the
filter\'s response decreases continuously. Although you can make this transition arbitrarily sharp
(at the cost of filter complexity), it can never be instantaneous, at least not in filters physically
realizable with today\'s technology.
The Bode and Phase Plots
Bode plots describe the behavior of a filter by relating the magnitude of the filter\'s response
(gain) to its frequency. An example of this type of plot is shown in Figure 2.
Figure 2. Filter responses are plotted on Bode plots, wh.
Description about -
1.filter and types of filter
2.comb filter defination & its magnitude and phase response
3.Digital comb filter defination & its magnitude and phase response
4.Digital comb filter using a digital differentiator & its magnitude and phase response
5.Comb Filters with Multiple Delay Elements defination & its magnitude and phase response
6.Digital Integrator(non-delaying) defination & its magnitude and phase response
7.Delaying Integrator defination & its magnitude and phase response
8.Fourier Transform & Dirac Delta Function (Unit Impulse Response) and its properties
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
1. Active Filters
Introduction, Active versus Passive Filters,
Types of Active Filters, First-Order Filters,
The Biquadratic Function, Butterworth
Filters, Transfer Function Realizations, Low
pass Filters, High-Pass Filters, Band-Pass
Filters, Band-Reject Filters, All-Pass Filters,
Switched Capacitor Filters, Filter Design
Guide Lines.
Syllabus
2. Filter Basics
• A filter is a frequency-selective circuit that passes
a specified band of frequencies and blocks or
attenuates signals of frequencies outside this band.
• A filter is used to remove (or attenuate) unwanted
frequencies in an audio signal
• “Stop Band” – the part of the frequency spectrum
that is attenuated by a filter.
• “Pass Band” – part of the frequency spectrum
that is unaffected by a filter.
• Filters are usually described in terms of their
“frequency responses,” e.g. low pass, high pass,
band pass, band reject (or notch)
3. Advantages of Active Filters over Passive Filters
(i) The maximum value of the transfer function or gain is greater
than unity.
(ii) The loading effect is minimal, which means that the output
response of the filter is essentially independent of the load
driven by the filter.
(iii) The active filters do not exhibit insertion loss. Hence, the
passband gain is equal to 0 dB.
(iv) Complex filters can be realized without the use of inductors.
(v) The passive filters using R, L and C components are
realizable only for radio frequencies, because the inductors
become very large, bulky and expensive at audio
frequencies. Due to low Q at low frequency applications,
high power dissipation is incurred. The active filters
overcome these problems.
4. (vi) Rapid, stable and economical design of filters for variety of
applications is possible.
(vii) The active filters are easily tunable due to flexibility in gain
and frequency adjustments.
(viii) The op-amp has high input impedance and low output
impedance. Hence, the active filters using op-amp do not
cause loading effect on the source and load. Therefore,
cascading of networks does not need buffer amplifier.
(ix) Active filters for fixed frequency and variable frequency can
be designed easily. The adjustable frequency response is
obtained by varying an external voltage signal.
(x) There is no restriction in realizing rational function using
active networks.
(xi) Use of active elements eliminates the two fundamental
restrictions of passivity and reciprocity of RLC networks.
5. Limitations of Active Filters over Passive Filters
(i) The high frequency response is limited by the gain-
bandwidth product and slew rate of the practical
op-amps, leading to comparatively lower
bandwidth than the designed bandwidth.
(ii) The design of active filters becomes costly for high
frequencies.
(iii) Active filters require dual polarity dc power
supply whereas passive filters do not.
(iv) The active element is prone to the process
parameter variations and they are sensitive to
ambient conditions like temperature. Hence, the
performance of the active filter deviates from the
ideal response.
8. 8
Types of Filters
• Butterworth – Flat response in the pass
band & stop band and called flat-flat filter.
• Chebyshev – steeper roll-off but exhibits
pass band ripple (making it unsuitable for
audio systems) & flat stop band.
• Cauer – It has equiripple both in pass &
stop band.
10. Filters Based On Frequency
Low pass filter (LPF) High pass filter (LPF)
20db/decade
20db/decade
11. Understanding Poles and Zeros
The transfer function provides a basis for determining
important system response characteristics
The transfer function is a rational
function in the complex variable s = σ + jω, that is
zi’s are the roots of the equation N(s) = 0, and are defined to be
the system zeros, and the pi’s are the roots of the equation
D(s) = 0, and are defined to be the system poles.
N(s) = 0; Zeros. D(s) = 0; Poles.
12. Example
A linear system is described by the differential equation
Find the system poles and zeros. Solution: From the
differential equation the transfer function is
Zero at -1/2
Poles at -2 & -3
14. • Consider a Pole at Zero. Its response is constant.
• Consider Poles at +a and –a. The exponential responses
are shown, for a function k/s+a, and k/s-a
• Consider conjugate poles +jω & -jω & their mirror image
on the right side, along with their responses which is
decaying sine wave and increasing sine wave.
15. The equation shown has 3 poles & one Zero at -1.
Zeros show how fast the amplitudes vary.
16. Frequency Response of filters
• Ideal
• Practical
• Filters are often described in terms of poles and
zeros
– A pole is a peak produced in the output spectrum
– A zero is a valley (not really zero)
18. Comparison of FIR & IIR Filter
1. FIR (Finite Impulse Response) (non-
recursive) filters produce zeros.
2. In signal processing, a finite impulse response
(FIR) filter is a filter whose impulse response (or
response to any finite length input) is
of finite duration, because it settles to zero in finite
time.
3. Filters combining both past inputs and past outputs
can produce both poles and zeros.
4. FIR filters can be discrete- time or continuous-time,
and digital or analog.
5. FIR filters are dependent upon linear-phase
characteristics.
6. FIR is always stable
7. FIR has no limited cycles.
8. FIR has no analog history.
9. FIR is dependent upon i/p only.
10. FIR’s delay characteristics is much better, but
they require more memory.
11. FIR filters are used for tapping of a higher-
order.
1. IIR (infinite Impulse Response) (recursive)
filters produce poles.
2. This is in contrast to infinite impulse response (IIR)
filters, which may have internal feedback and may
continue to respond indefinitely (usually decaying).
3. IIR filters are difficult to control and have no
particular phase.
4. IIR is derived from analog.
5. IIR filters are used for applications which are not
linear.
6. IIR can be unstable
7. IIR filters make polyphase implementation possible.
8. IIR filters can become difficult to implement, and also
delay and distortion adjustments can alter the poles
& zeroes, which make the filters unstable.
9. IIR filters are dependent on both i/p and o/p.
10. IIR filters consist of zeros and poles, and require less
memory than FIR filters.
11. IIR filters are better for tapping of lower-orders, since
IIR filters may become unstable with tapping higher-
orders.
19. ACTIVE FILTERS USING OP-AMP:
Filters are frequency selective circuits. They are required to pass
a specific band of frequencies and attenuate frequencies outside
the band. Filters using an active device like OPAMP are called
active filters. Other way to design filters is using passive
components like resistor, capacitor and inductor.
ADVANTAGES OF ACTIVE FILTERS:
Possible to incorporate variable gain
Due to high Zi & Z0 of the OPAMP, active filters do not load the
input source or load.
Flexible design.
20. FREQUENCY RESPONSE OF FILTERS:
Gain of a filter is given as, G=Vo/Vin
Ideal & practical frequency responses of different types of
filters are shown below.
22. Because of simplicity, Butterworth filters are considered.
• In 1st. order LPF which is also known as one pole
LPF. Butterworth filter and it’s frequency response
are shown above.
• RC values decide the cut-off frequency of the filter.
• Resistors R1 & RF will decide it’s gain in pass band.
As the OP-AMP is used in the non-inverting
configuration, the closed loop gain of the filter is given
by
1
1
R
R
A F
VF
23. )1(1
in
C
C
V
jXR
jX
V
fC
XC
2
1
)2(
2
12
2
1
2
1
1
j
fRC
V
jfRC
jV
fC
jR
V
fC
j
V inin
in
fRCj
Vin
21
EXPRESSION FOR THE GAIN OF THE
FILTER:
Reactance of the capacitor is,
Equation (1) becomes
Voltage across the capacitor
V1 =
24. f = frequency of the input signal
H
VF
in
inF
VF
f
f
j
A
V
V
fRCj
V
R
R
VAV
1
21
1
0
1
10
Output of the filter is,
25. The operation of the low-pass filter can be verified from
the gain magnitude equation, (7-2a):
1. At very low frequencies, that is, f < fH,
2. At f = fH,
3. f < fH,
26. DESIGN PROCEDURE:
Step1: Choose the cut-off frequency fH
Step2: Select a value of ‘C’ ≤ 1µF (Approximately
between .001 & 0.1µF)
Step3: Calculate the value of R using
Step4: Select resistors R1 & R2 depending on the desired
pass band gain. (Try different gains)
=2. So RF=R1
27. For a first order Butterworth LPF, calculate the cut –off
frequency if R=10K & C=0.001µF.Also calculate the
pass band voltage gain if R1=10K RF =100K
KHz
RC
fH 915.15
10001.010102
1
2
1
63
Design a 1st order LPF for the following specification
Pass band voltage gain = 2. Cut off frequency, fC = 10KHz.
AVF = 2; Let RF = 10K
1+100K/10K =11
RF/R1=1 Let C = 0.001µF
63
10001.010102
1
2
1
&
2
1
Cf
R
RC
f
H
H
R=15.9K
28. Circuit diagram & frequency response are shown
above.
Again RC components decide the cut off frequency
of the HPF where as RF & R1 decide the closed loop
gain.
1st ORDER HPF:
fL is shown for HPF
30. SECOND-ORDER LOW-PASS BUTTERWORTH FILTER
The gain of the second-order filter is set by R1, and RF,
while the high cutoff frequency fH is determined by R2,
C2, R3, and C3, as follows:
32. Filter Design
1. Choose a value for the high cutoff frequency fH
2. To simplify the design calculations, set R2 = R3 = R and
C2 = C3 = C. Then choose a value of C ≤ 1µF
3. Calculate the value of R using Equation for fH:
4. Finally, because of the equal resistor (R2 = R3) and
capacitor (C2 = C3) values, the pass band voltage gain AF
= (1 + RF/R1) of the second-order low-pass filter has to be
equal to 1.586. That is, RF = 0.586/R1 This gain is
necessary to guarantee Butterworth response. Hence
choose a value of R1 < 100 kΩ and calculate the value of
RF .
33. As in the case of the first-order filter, a second-order
high-pass filter can be formed from a second-order
low-pass filter simply by interchanging the
frequency determining resistors and capacitors.
Figure 7-8(a) shows the second-order high-pass
filter.
SECOND-ORDER HIGH-PASS BUTTERWORTH FILTER
𝒗 𝒐
𝒗𝒊𝒏
=
𝑨 𝑭
𝟏 +
𝒇 𝑳
𝒇
𝟒
AF = 1.586 for 2nd order
Butterworth Filter
36. Writing Kirchhoff's current law at node VA(S),
I1 = I1 + I2.
we have omitted S; for example Vin(S) is written as Vin.
Also, using the voltage-divider rule,
since RiF = ∞, IB = 0 A
Substituting the value of VA in Equation (C-7) and
solving for Vh we get
C-7
37. where AF = 1 + (RF/R1)-Therefore,
Solving this equation for V0/Vin, we have
38. For frequencies above fH, the gain of the second-order low-
pass filter rolls off at the rate of -40dB/decade. Therefore,
the denominator quadratic in the gain (V/Vin) equation must
have two real and equal roots. This means that
41. Bi-Qadratic Function
ωo = undamped natural (or resonant) frequency
Q = quality factor or figure of merit
K = DC gain
Substituting ko =1, k1=0, k2=0, for Lowpass Filter
Second-Order Low-Pass Filters
42. A first-order filter can be converted to a second-
order filter by adding an additional RC network,
known as the Sallen-Key circuit.
The transfer function of the filter network is
12-47
12-46
43. Comparing the denominator of Eq. (12.50) with that of
Eq. (12.46) shows that Q can be related to K by
44. The frequency response of a second-order system at the 3-
dB point will depend on the damping factor ζ such that Q
= 1/ 2ζ (zeta). A Q-value of ( = 0.707), which represents
a compromise between the peak magnitude and the
bandwidth, causes the filter to exhibit the characteristics of
a flat passband as well as a stop band, and gives a fixed
DC gain of K = 1.586:
𝟏
𝟐
However, more gain can be realized by adding a voltage-
divider network. as shown in Fig. 12.14, so that only a
fraction x of the output voltage is fed back through the
capacitor C2 that is,
45. Thus, for Q - 0.707, xk is - 1.586, allowing a designer to
realize more DC gain K by choosing a lower value of x,
where x < 1.
Fig:12.14
46. Example:12:3
Solution
To simplify the design calculations, let R1 = R2 = R3 =R4
= R and let C2 = C3 = C. Choose a value of C less than or
equal to 1 µF. Let C = 0.01 µF. For R2 = R3 = R and C2 =
C3 = C, Eq. (12.49) is reduced to
RF = (K - 1)R1 = (4 - 1) = x = 15,916 = 47,748
50. Second-Order High-Pass Filters
The transfer function can be derived by applying the RC-
to-CR transformation and substituting 1/s for s in Eq.
(12.47). For R1 = R2 = R3 = R, and C2 = C3 = C, the
transfer function becomes
56. DESIGN EQUATIONS:
Select C1 = C2 =C
FCCAf
Q
R
2
1
FC AQCf
Q
R
22
22 Cf
Q
R
C
B
A is the gain at f =fC
1
3
2R
R
AF
Condition on gain AF<2Q2
61. ALL PASS FILTER:
It is a special type of filter which passes all the
frequency components of the input signal to output
without any attenuation. But it introduces a
predictable phase shift for different frequency of
the input signal.
63. Switched-Capacitor Filters
• Active RC filters are difficult to implement totally on
an IC due to the requirements of large valued capacitors
and accurate RC time constants
• The switched capacitor filter technique is based on the
realization that a capacitor switched between two
circuit nodes at a sufficiently high rate is equivalent to a
resistor connecting these two nodes.
• Switched capacitor filter ICs offer a low cost high order
filter on a single IC.
• Can be easily programmed by changing the clock
frequency.
66. Zeros: roots of N(s)
• Poles: roots of D(s)
• Poles must be in the left half plane for the system to be stable
• As the poles get closer to the boundary, the system becomes less stable
• Pole-Zero Plot: plot of the zeros and poles on the complex s plane
H(s) =
𝑵(𝒔)
𝑫(𝒔)
X
X
X
X X
X
X
X
X
-a +a
RealImaginary
jω
-jω
𝒌
𝒔 + 𝒂
𝒌
𝒔 − 𝒂