Zoom Fft free essay sample

A Seminar Report On ZOOM FFT Submitted In partial fulfillment For the award of the Degree of Bachelor of Technology Applied Electronics amp; Instrumentation In Department of Electronics amp; Communication Engineering Submitted To: Submitted By Mr. Raj Kumar Jain Krishan Gopal Bansal HOD Enrollment No: 9E1CIAIM40P021 Department of Electronics amp; Communication Engineering CompuCom Institute of Information Technology amp; Management Rajasthan Technical University, Kota April 2013 ACKNOWLEDGEMENT The seminar has been a unique experience for me instead of routine and momentary exercise. It has leap to new field of acquiring knowledge and learning. First of all I wish to express my sincere thanks to the Rajasthan Technical University, Kota. This introduces the scheme of providing seminar for technical student during for Technical student during the 4-year course of B. Tech. With the drastic development of technology amp; speedy industrializations of the country I consider myself to fortunate to have undergone seminar on zoom FFT Techniques in Blood Flow Analysis. We will write a custom essay sample on Zoom Fft or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page I am very thankful to Mr. Manvendra Singh amp; Mr. Dilip Tiwari sir or other concerned Person for their guidance, constant encouragement; strong support amp; kind help of  Understand many technical aspects in my training period. My heartily thanks to Mr. Raj Kumar Jain ,HOD (Electronics amp; Communication Department) and of CIITM, JAIPUR for all kind of help they have granted in absence of which the seminar would have not been possible. Krishan Gopal Bansal Enrollment No: 9E1CIAIM40P021 PREFACE The seminar is an essential requirement for an engineering student. The student has to give the seminar for the pre described period as per the university norms. The purpose of seminar is to help the student to gain industrial experience. Moreover, as for the utility of seminar concerning, it can be said that student gets an opportunity during his training to imply the theoretical knowledge in the field work and to clear the difficulties in a better way. In the year 2013, in my 8th semester, I give my seminar on ZOOM FFT Technique in Blood Flow Analysis under the guidance of Mr. Manvendra Singh. Guided BySubmitted By Mr. Manvendra Singh Krishan Gopal Bansal Enrollment No: 9E1CIAIM40P021 Submitted To Mr. Raj Kumar Jain HOD (EC/ AEI) ABSTRACT I give my seminar on zoom FFT techniques in Blood Flow analysis. According to the rule of  Rajasthan Technical University towards the fulfillment of four years degree course of B. Tech. The object of seminar in engineering field is to co-relate the theory with practical knowledge and to make student familiar with industrial environment. I have been fortunate to get seminar in such an industry which has been continuous by climbing the ladder of development utilizing the latest technology. Here I could get knowledge of  various equipment which are tested amp; calibrated in their enterprise. Contents Page No. Certificate i Acknowledgement ii Preferenceiii Abstract iv List of Figures vii Chapter 1: Introduction1 1. 1 Doppler Effect Phenomenon2 1. 2 Real Blood Flow Analysis4 Chapter 2: Down Sampling6 2. 1 Maintaining the Sampling Theorem Criterion6 2. 2 Down Sampling Process7 2. 3 Down Sampling by Rational Fraction7 Chapter 3: Blood Flow9 3. 1 Cell –Tissue-Organ-System9 3. 2 Blood and Its Composition9 3. 3 The Mechanics of Blood Circulation9 3. 4 The Basics of Motion10 3. 5 Basic Ideas in Fluid Mechanics12 Chapter 4: Fast Fourier Transform14 4. 1 Definition and Speed15 Chapter 5: ZOOM FFT16 5. 1 Basic Principle of The Zoom FFT18 5. Zoom FFT Algorithm19 5. 3 Simulation Result25 5. 4 Advantages28 5. 5 Applications28 Conclusion29 References30 List of Figure Figure NamePage No. 1. Doppler Effect3 2. Formal FFT4 3. Zoom FFT5 4. Ultrasonic Signal Path5 5. Blood flow graph with velocity12 6. Zoom FFT analysis 117 7. Zoom FFT analysis 218 8. Schematic diagram of the Zoom FFT process19 9. Digital Mixing20 10. Zoom FFT Algorithm20 11. Hilbert Transform22 12. FFT of a wave with 2 frequencies23 13. Zoom FFT with various Wave form24 14. Simulation results without zoom25 15. Simulation results with zoom26 16. Simulation results with zoom/without zoom and input signals27 Chapter 1 Introduction An adequate blood flow supply is necessary for all organs of the body. Analysis of the blood flow finds its importance in the diagnoses of diseases. There are many techniques for analyzing the blood flow. These techniques are not affordable by the poor people because of their high expense. So we have implemented a technique called Zoom-FFT. This technique is simple and affordable to detect the blood clots and other diseases. Human with his potential tries to get whichever is unexplored, explored, and till now we are managing and succeeding using some technical ways. In the same way this is one of the explorations made for scanning the intra details of some specific objects using ultrasound named SONOGRAPHY, which is used as an alternative to x-ray photography. In this paper, the method to zoom the image or the scanned data-using zoom FFT has been discussed. It also explains the algorithm to get ZOOM FFT and how it can be obtained via simulation. Real time experimentation and its applications, with basics of ultrasound scanning are also explained. Here a specific application will be dealt i. e. , ultrasonic blood flow analyzer using ZOOM FFT. Blood flow analysis is done by passing a high frequency ultrasonic wave in the blood vessels through a transducer (transmitter) . The reflected signal; from the receiver transducer has a different frequency due to the Doppler principle. This signal is passed to a DSP processor to find the frequency spectrum. Because of the high frequency of the ultrasonic wave, the resolution of the frequency spectrum output will not be good. Therefore we go for advanced Zoom FFT technique, wherein a very small frequency change due to the clot formation can be obtained with a good resolution. It can be used to locate the initial presence of a blood clot. All of these tasks must be achieved with a single DSP chip in order for the system to be both cost-effective and power efficient and thus widely accepted. This seminar report proposes: 1. Study of Bio-medical signal processing 2. Mixing down the input signal to the base band frequency using Hilbert Transform 3. Finding the down sampling using the decimation process 4. Obtaining the spectrum output using fast Fourier transform 5. Simulation is done by Matlab/C. 6. TMS320C5X/6X DSP processor does real time implementation. SOUND IS A COMPRESSIONAL WAVE† Sounds at frequencies above the audible range, to say above 20 KHz are Ultrasonic wave, in the megahertz range. Above which are supersonic sound. 1. 1 DOPPLER EFFECT PHENOMENON A shift in frequency (f) of the wave will be expected due to the source and observers motion relative to each other. If the distance between them is reduced or increased. That shift in frequency depends on the velocity of sound which also depends on density of the medium, in which it propagates. When a small object is situated in the path of the sound wave, the wave will be resisted (scattered). A direct measurement of this velocity will provide useful information about the dynamic property of the medium. The Velocity of sound in Blood is 1570 m/s. Perceived velocity is V’=V-V0 In terms of frequency (f), as a velocity dependent factor. Fp = f0 (V+V0)/V-Vs, for both objects moving towards. – (1) Fp = f0 (V-V0)/V+Vs, for both objects moving away from each other. – (2) F0: Actual Frequency. Fp: Perceived Frequency. V: Velocity of Wave. Vs: Source Velocity. V0: Velocity of Observer. Thus we get the perceived frequency proportionately changed with respect to changes in measuring media. This process is explained using animation as below in FIG (1). Fig. 1 Doppler Effect The Doppler Effect can be explained with respect to pitch or wavelength, since all are dependent to each other. E. g. of Doppler Effect: Say, A car passes you on the street blowing its horn at a frequency of 440Hz, the whole way, As the car approaches you, you will hear a pitch gt; 440Hz(in increasing order). After the car passes you and drives away from you, you will hear a pitch lower lt; 440Hz (in descending order). â€Å"THIS CONCEPT IS APPLIED IN ULTRASOUND RANGE FOR HUMAN BLOOD FLOW ANALYSIS USING VELOCITY OF BLOOD† Steps involved: Sound generation: The ultrasonic sound is generated using the piezoelectric transducer. * Number of transducer may vary from 1 to many. * Narrow beam of wave is to be feed in. * Continuous mode of operation with no timed switching is applied in real time to measure Frequency and Amplitude * Doppler shift analysis for frequency content is to be done. * Creation of image – to plot in 2 Dimension. * Display using color differentiation. 1. 2 REAL BLOOD FLOW ANALYSIS: In an Ultrasonic blood flow analysis, a beam of ultrasonic energy is directed through a blood vessel at a shallow angle and its transit time is then measured. More common are the ultrasonic analyzers based on the Doppler principle. An oscillator, operating at a frequency of several Mega Hertz, excites a piezoelectric transducer. This transducer is coupled to the wall of an exposed blood vessel and sends an ultrasonic beam with a frequency F into the flowing blood. A small part of the transmitted energy is scattered back and is received by a second transducer arranged opposite the first one as shown in. Fig. 2 Formal FFT Fig. 3 Zoom FFT Fig 4 Ultrasonic Signal Path Chapter 2 Down sampling In signal processing,  down sampling  (or sub sampling) is the process of  reducing the sampling rate  of a  signal. This is usually done to reduce the  data rate  or the size of the data. The down sampling factor (commonly denoted by  M) is usually an integer or a rational fraction greater than unity. This factor multiplies the sampling time or, equivalently, divides the sampling rate. For example, if  compact disc audio  at 44,100  Hz is down sampled to 22,050  Hz before broadcasting over  FM radio, the  bit rate  is reduced in half, from 1,411,200 bit/s to 705,600 bit/s, assuming that each sample retains its bit depth of 16 bits. The audio was therefore down sampled by a factor of 2. 2. 1 Maintaining the sampling theorem criterion Since down sampling reduces the sampling rate, we must be careful to make sure the  Shannon-Nyquist sampling theorem  criterion is maintained. If the sampling theorem is not satisfied then the resulting digital signal will have  aliasing. To ensure that the sampling theorem is satisfied, a  low-pass filter  is used as an  anti-aliasing filter  to reduce the bandwidth of the signal  before  the signal is down sampled; the overall process (low-pass filter, then down sample) is called  decimation. Note that if the original signal had been bandwidth limited, and then first sampled at a rate higher than the  Nyquist minimum, then the down sampled signal may already be Nyquist compliant, so the down sampling can be done directly without any additional filtering. Down sampling only changes the sample rate not the bandwidth of the signal. The only reason to filter the bandwidth is to avoid the case where the new sample rate would become lower than the Nyquist requirement and then cause the aliasing by being below the Nyquist minimum. Thus, in the current context of down sampling, the anti-aliasing filter must be a low-pass filter. However, in the case of sampling a  continuous signal, the anti-aliasing filter can be either a low-pass filter or a  band-pass filter. A band pass signal, i. e. a band-limited signal whose minimum frequency is different from zero, can be down sampled avoiding superposition of the spectra if certain conditions are satisfied. 2. 2 Down sampling process Consider a  discrete signal  Ã‚  on a radian frequency  digital frequency  range. Down sampling by integer factor Let  M  denote the down sampling factor. 1. Filter the signal to ensure that the sampling theorem is satisfied. This filter should, theoretically, be the  sinc filter  with frequency cutoff at. Let the filtered signal be denoted. 2. Reduce the data by picking out every  Ã‚  sample:. Data rate reduction occurs in this step. The first step calls for the use of a perfect low-pass filter, which is not implementable for real-time signals. When choosing a realizable low-pass filter this will have to be considered along with the aliasing effects it will have. Realizable low-pass filters have a skirt, where the response diminishes from near unity to near zero. So in practice the cutoff frequency is placed far enough below the theoretical cutoff that the filters skirt is contained below the theoretical cutoff. 2. 3 Down sampling by rational fraction Let  M/L  denote the down sampling factor. 1. Up sample  by a factor of  L 2. Down sample by a factor of  M Note that a proper up sampling design requires an interpolation filter after increasing the data rate and that a proper down sampling design requires a filter before eliminating some samples. These two low-pass filters can be combined into a single filter. Also note that these two steps are generally not reversible. Down sampling results in a loss of data and, if performed first, could result in data loss if there is any data filtered out by the down samplers low-pass filter. Since both interpolation and anti-aliasing filters are low-pass filters, the filter with the smallest bandwidth is more restrictive and can therefore be used in place of both filters. When the rational fraction  M/L  is greater than unity then  Ã‚  and the single low-pass filter should have cutoff at. NOTE: Up sampling first is necessary in all cases where the rate is not an even multiple. E. g. : if a sample rate of 2 xs is changed to a rate of 1x by averaging every pair of samples this would be equivalent to a low pass filtering operation. But taking every other sample would be equivalent to up then down sampling in this special case where the multiple was 2 to 1, so there is no need to do an up sample first. Chapter 3 Blood flow Blood flow  is the continuous running of  blood  in the  cardiovascular system. The human body is made up of several processes all carrying out various functions. We have the  gastrointestinal system  which aids the  digestion  and the  absorption  of food. We also have the  respiratory system  which is responsible for the absorption of  O2  and elimination of  CO2  . The  urinary system  removes waste from the body. The  cardiovascular system  helps to distribute food,  O2  and other product of  metabolism. The  reproductive system  is responsible for perpetuating the species. The  nervous and  endocrine system  is responsible for coordinating the integration and function of other system. 3. 1 Cell –Tissue-Organ-System The  cell  is the basic structure in the human body. These  cells  that makes up the bodies of all living things exist in an ‘internal sea’ of  extracellular  fluid (ECF) enclosed within the integument of the animal. From this  fluid, the  cell  takes up  O2  and  nutrients  into it, they discharge  metabolic  waste products. In animals with a closed  vascular  system, the  ECF  is divided into two components, the interstitial  fluid and the circulating  blood  plasma. The  plasma  and the  cellular  elements of the  blood, principally  red blood cells, fill the  vascular system  and together they constitute the total  blood volume. 3. 2 Blood and Its composition Blood  is the  viscous  fluid composed of  plasma  and  cells. The composition of the blood includes plasma,  red blood cells,  white blood cells  and  platelets. In the  microcirculation  the properties of the blood cells have an important influence on flow. 3. 3 The mechanics of blood circulation Mechanics  is the study of motion (or equilibrium) and the  forces  that cause it. The blood moves in the blood vessels, while the  heart  serves as the pump for the blood. The vessel walls of the heart are elastic and are movable, therefore causing the blood and the wall to exert forces on each other which in turn influence their respective motion. Therefore to understand the mechanics of circulation of the heart, it will be worth the while to go through a review of basic mechanics of fluid, and  elastic  solids (momentum) and the nature of the forces exerted between two moving substances in contact. 3. 4 The basics of motion The study of motion is born from the argument that there is no change in motion without  force. These beliefs were somewhat obscured until the seventeenth century when  Isaac Newton  formulated his three laws of motion. Another quantity of interest in describing the motion of a particle is its  velocity. The velocity basically is the rate of change of the position of an object with time. Blood velocities in  arteries  are higher during  systole  than during  diastole. One parameter to quantify this difference is  plasticity index  (PI), which is equal to the difference between the peak systolic velocity and the minimum diastolic velocity divided by the mean velocity during the  cardiac cycle. The rate of change of  position  as we saw is the  velocity. While the rate of change of velocity is referred to as the  acceleration. For a motion along a line, the acceleration is given as: This is the same as the slope of the tangent of the graph v against t. The unit of acceleration is meter per second’s squared (ms-2). Newton’s first law: every particle continues in a state of rest of uniform motion in a straight line unless acted on by some external force or forces. In other words the  velocity  remains constant (zero acceleration) if no force is acting on the body. Newton’s second law: when a particle of mass m is acted on by a force so that it experiences an  acceleration  a, the net force acting on it is equal to the mass multiplied by the  acceleration. That is to say the net force is a vector which we can call F. which is given by F= ma. This equation is an important equation in mechanics. It is called the equation of motion of a particle. The term net force means the sum of all forces acting on the particle, which may be exerted in different ways. Newton’s third law: Newton’s third law has commonly been described as action and reaction. Basically the law is defined as, to every action there is equal and opposite reaction. That is to say, if one body exerts a force F on another body, the second body must also exerts an equal and opposite reaction on the formal body (-F). The negative on the force by no means suggest that force can be negative, the negative only means that the force is acting on the opposite direction, it has nothing to do with the magnitude it is simply the direction. As an illustration if you press a stone to your finger, unknown to you, your finger exerts a force also on the stone which is equal in  magnitude. Momentum We recall from Newton’s second law that F= ma. When expanded the above can be written as . The quantity mv is called the momentum of a particle. Therefore we can also say that Newton’s law from the equation above can be defined as force and is equal to the rate of change of momentum. In the absence of external forces the momentum of a particle of a body or system of particle remains constant or is conserved. What we mean is that, if two particles (masses m1 and m2) and velocities v_1 and v_2 collides, the combined body (m_1 + m_2 ) must have the same momentum as the original body put together, so its velocities must be (m_1v_1 + m_2v_2)/ m_1 +m_2. If the collision is instantaneous this would be true immediately. Thus we also have inelastic and elastic collision depending on the relative motion of the masses, or continuous. 3. 5 Basic ideas in fluid mechanics Basically the force experienced by fluids includes long range and short range. The long range force includes gravitational and electromagnetic forces. The electromagnetic force on an element depends on the quantities like its electric charge, but on the other hand the gravitational force depends only on its mass. We will consider from this point the gravitational force alone. If we have a fluid with element p which occupies the point x at a certain time t and has a volume v and if the fluid in the neighborhood of x, at that time has a density ? then the gravitational force on the element is given as: Fig. 5 Blood flow graph with velocity Laminar shear of fluid between two plates. . Friction between the fluid and the moving boundaries causes the fluid to shear (=flow). The force required for this action per unit area is the stress. The relation between the stress (=force) and the shear rate (=flow velocity) determines the viscosity. In a stationary fluid the only stress present is the  pressure. Let us assume a body force that is vertical; we can assume the horizontal component of the pressure force must balance out. Pressure varies with height as we know, as we move the height the pressure changes. If A is the horizontal cross-sectional area of the element and if p1 and p_2 are the pressure on the surface, where p1 is the pressure  on the bottom and p2 is the pressure on the top, then the net upward force on the element can be given as (p_1 – p_2 )A This must be equal to the weight which acts downwards and is equal to the density of the fluid times its volume ( AZ’) where Z’ is the depth of the element times g. p_1 – p_2 )A = g? AZ’ (p_1 – p_2 ) = g? Z’ If we assume that the  pressure  is  atmospheric  and assuming Z=0 that is to say Z is negative in the fluid. While p is given as P_1 = p_2 g? Z The above equation there is the called the  hydrostatic pressure The magnitude of the viscous stresses depends upon the rate of deformation. For example when a body is moved rapidly through a fluid it causes more rapid deformation of fluid element than one moving slowly. Viscosity is giving as where mu is a constant called the coefficient of viscosity. Chapter 4 Fast Fourier transform A  fast Fourier transform  (FFT) is an  algorithm  to compute the  discrete Fourier transform  (DFT) and it’s inverse. There are many different FFT algorithms involving a wide range of mathematics, from simple  complex-number arithmetic  to  group theory  and  number theory; this article gives an overview of the available techniques and some of their general properties, while the specific algorithms are described in subsidiary articles linked below. The DFT is obtained by decomposing a  sequence  of values into components of different frequencies. This operation is useful in many fields (see  discrete Fourier transform  for properties and applications of the transform) but computing it directly from the definition is often too slow to be practical. An FFT is a way to compute the same result more quickly: computing the DFT of  N points in the naive way, using the definition, takes  O(N2) arithmetical operations, while an FFT can compute the same DFT in only O(N  log  N) operations. The difference in speed can be enormous, especially for long data sets where  N  may be in the thousands or millions. In practice, the computation time can be reduced by several  orders of magnitude  in such cases, and the improvement is roughly  proportional to  N  / log (N). This huge improvement made the calculation of the DFT practical; FFTs are of great importance to a wide variety of applications, from  digital signal processing  and solving  partial differential equations  to algorithms for quick  multiplication of large integers. The best-known FFT algorithms depend upon the  factorization  of  N, but there are FFTs with O (N  log  N)  complexity  for all  N, even for  prime  N. Many FFT algorithms only depend on the fact that  Ã‚  is an  Nth  primitive root of unity, and thus can be applied to analogous transforms over any  finite field, such as  number-theoretic transforms. Since the inverse DFT is the same as the DFT, but with the opposite sign in the exponent and a 1/N  factor, any FFT algorithm can easily be adapted for it. The FFT has been described as the most important  numerical algorithm  of our lifetime 4. 1 Definition and speed An FFT computes the  DFT  and produces exactly the same result as evaluating the DFT definition directly; the only difference is that an FFT is much faster. In the presence of  round-off error, many FFT algorithms are also much more accurate than evaluating the DFT definition directly, as discussed below. ) Let  x0 xN-1  be  complex numbers. The DFT is defined by the formula Evaluating this definition directly requires O(N2) operations: there are  N  outputs  Xk, and each output requires a sum of  N  terms. An FFT is any method to compute the same results in O (N  log  N) operations. More precisely, all known FFT algorithms require  ? (N  log  N) operations (technically, O only denotes an  upper bound), although there is no known proof that a lower complexity score is impossible. To illustrate the savings of an FFT, consider the count of complex multiplications and additions. Evaluating the DFTs sums directly involves  N2  complex multiplications and  N(N? 1) complex additions [of which O(N) operations can be saved by eliminating trivial operations such as multiplications by 1]. The well-known radix-2  Cooley–Turkey algorithm, for  N  a power of 2, can compute the same result with only (N/2)log2(N) complex multiplies (again, ignoring simplifications of multiplications by 1 and similar) and  Nlog2(N) complex additions. In practice, actual performance on modern computers is usually dominated by factors other than the speed of arithmetic operations and the analysis is a complicated subject (see, e. g. , Frigo amp; Johnson, 2005), but the overall improvement from O(N2) to O(N  log  N) remains. Chapter 5 ZOOM FFT The Zoom-FFT is a process where an input signal is mixed down to baseband and then decimated, prior to passing it into a standard FFT. The advantage is for example that if you have a sample rate of 10 MHz and require at least 10Hz resolution over a small frequency band (say 1 KHz) then you do not need a 1 Mega point FFT, just decimate by a factor of 4096 and use a 256 point FFT which is obviously quicker. In contrast, the zoom-FFT uses digital down conversion techniques to localize the standard FFT to a narrow band of frequencies that are centered on a higher frequency. The zoom-FFT is used to reduce the sample rate required when analyzing narrowband signals E. G. in HF communications. Zoom FFT analysis is simply an efficient computation of a subset of the FFT. You use this kind of tool when you are mainly interested in a certain frequency band of 10 kHz to 11 kHz. Rather than computing the FFT for the entire frequency range, you only perform computations on a subset of frequencies. Thus, you can save a significant amount of processing power and time using this method. Zoom FFT Analysis is a technique that provides better resolution on FFT analysis allowing the user to select a start and ending frequency with a given set of FFT lines such as 1600 or 3200. Since the start and end frequency can be arbitrarily selected then very good resolutions can be achieved in narrow spans within a baseband frequency span. WHY TO ZOOM? Minute variations in blood flow can be seen E. G. (starting stage of blood clot) the variation in the blood flow via, the zoom FFT will be more evident practically. This may be in frequency domain or can be imaged in 2D for â€Å"VISUAL PERCEPTION†. Normal blood vessel f = f+ df Blood vessel with clot formation f= f+df+Df Fig 6 Zoom FFT analysis Zoom FFT analysis is used to concentrate, or â€Å"zoom,† the FFT analysis on a narrow band of frequencies. This improves the frequency resolution and helps you to distinguish closely-spaced frequencies. As for the standard FFT analysis, the frequency resolution is related to the total acquisition time. So, to achieve better frequency resolution, longer acquisition time is required. Fig 7 Zoom FFT analysis This example shows how two closely-spaced frequencies are barely detectable when using the standard FFT analysis in baseband frequency span. Zooming on the appropriate frequency range (10 kHz here) then clearly reveals the two different frequencies. 5. 1 Basic principle of the Zoom FFT The zoom FFT (Fast Fourier Transform) is a signal processing technique used to analyses a portion of a spectrum at high resolution. Fig shows the spectrum of a real signal, with the region of interest shaded. The steps to apply the zoom FFT to this region are as follows: * Frequency translates to shift the frequency range of interest down to near 0 Hz (DC), as shown is Fig. * Low pass filter to prevent aliasing when subsequently sampled at a lower sample rate, see Fig 1. * Re-sample at a lower rate. * FFT the re-sampled data. Multiple blocks of data are needed to have an FFT of the same length. The resulting spectrum will now have a much smaller resolution bandwidth, compared to an FFT of non-translated data, as shown in Fig. 1. Fig. 8 schematic diagram of the Zoom FFT process Where the figures are show 8a) Original spectrum with region of interest shaded 8b) Spectrum after frequency translation 8c) Spectrum after low passes (anti-aliasing) filtering. 8d)  Spectrum after sub-sampling (decimation). 5. 2 Zoom FFT Algorithm If the two pass filtering were done as described, and then the FFT of 1024 points of the data is taken, then the spectrum from 0 to 1600Hz will be a zoomed view of the original region of interest. The frequency resolution of this zoomed spectrum will be 4Hz (4096/16) sixteen times finer than the original 64Hz (65536/1024). To demonstrate the complete process a data buffer of 20000 points has been created, it contains random noise along with two high frequency tones, close together, around 19  kHz. Here is the source code of the zoom FFT process. The zoom frequency is 19  kHz, two passes of the anti-alias filter, sub-sample process are then carried out. The filter coefficients are those calculated above, the sub-sample factor is 4, so the frequency resolution is improved from 128  Hz to 8  Hz. The following diagram shows the zoom process: Fig. 9 Digital Mixing While the following diagram shows the basic architecture of the Zoom-FFT: Fig. 10 Zoom FFT Algorithm Spectrum analyzers originally provided the zoom FFT to offer higher frequency resolution over a specific bandwidth, given the limitation of a small amount of on-board memory. With the zoom FFT, you can obtain a very fine frequency resolution (narrow band analysis) without computing the entire spectrum. The ability to increase the frequency resolution of a spectral measurement in part of a frequency range. Zoom can also apply to time domain (oscilloscope) measurements. Digital zoom (frequency domain) is usually implemented by multiplying the input signal with a sine and cosine at a new desired center frequency, and then low-pass filtering the data, followed by sampling rate reduction (decimation). In contrast, a visual zoom simply increases the size of the plot of data without adding any new information. In traditional FFT Spectrum Analyzers, zoom was implemented in hardware to get around the memory limitations of the processors, which made it impossible to economically perform large Fourier Transforms. However, as memory size and processor speed has increased, large FFTs are now economically possible. Fig (4) shows the difference between FFT and Zoom FFT. When you need to have high frequency resolution this can be achieved in a number of different ways: 1. Large FFT: Has the advantage that it gives the keeps all spectral lines over the entire frequency range, whereas zoom only picks a sub-set of a given frequency range. Thus with zoom, multiple computations must be made to cover a broader frequency range. 2. Destructive zoom: The traditional zoom method implemented with digital filters, which throw away frequency information outside the selected range. 3. Non-destructive zoom: A zoom technique, which keeps the entire original time function. Thus zoom can be performed in different frequency ranges on the same data without requiring the acquisition of new data. INPUT SIGNAL: The input signal for the frequency under design can be a cosine wave or a sine wave this periodic is only for the implementation of the work. For real time implementation any non-periodic signal can also be considered. FREQUENCY TRANSLATION: The signal, which is of high frequency, should be translated to a low frequency to get the proper response of the input signal. This is implemented by frequency translation. Fig 11 Hilbert Transform If cos (1900) is considered as an input signal it can be translated to cos (100) by the following procedure as depicted in the figure(4). The output arrived is as follows, cos(A-B) = cos(A) cos(B) + sin(A) sin(B) – (3) i. e, cos(2000-1900)=cos(2000) cos(1900) + sin(2000) sin(1900) = cos(100) Thus the input wave, which has frequency 1900Hz, is translated into 100Hz. Discrete Hilbert Transform formula The frequency translating function eq-4. f(n)=m=0N-11—1m-nfmcotm-n(? /N) – (4) Decimation Re sampling at discrete instances, the already sampled wave. As in eq-5, Ym=k=0N-1hkxMm-k, M=decimation factor. – (5) FFT: The Fast Fourier Transform (FFT) is an algorithm that efficiently contains the frequency domain conversion as in Fig (8) and (9). Xk=n=0N-1xne-j2? nk/N Fig. 12 FFT of a wave with 2 frequencies Fig. 13 Zoom FFT with various Wave form 5. 3 SIMULATION RESULTS Fig. 14 Simulation results without zoom Fig. 15 Simulation results with zoom Fig. 16 Simulation results with zoom/without zoom and input signals. 5. 4 ADVANTAGES 1. Increased frequency domain resolution 2. Reduced hardware cost and complexity . Wider spectral range In places where the frequency content has to be analyzed, this zooming FFT can be utilized, mainly for the hidden glitches during signal frequency transition. 5. 5 APPLICATIONS: * Ultrasonic blood flow analysis. * RF communications. * Mechanical stress analysis. * Doppler radar. * Bio-medical fields. * Side band analysis, and modulation analys is CONCLUSION Currently the seminar report has been tested on the simulation basis, the output of the simulations are satisfactory. Real time experimentation is being done, using the piezo electric ultrasonic transducer for verification purpose. The frequencies content from the media are obtained and currently they are being transformed to image as 2D for visual perception. In places where the frequency content has to be analyzed, this zooming FFT can be utilized, mainly for the hidden glitches during signal frequency transition. References 1. Edited by Lawrence R. Rabiner, Charles M. Rader, â€Å"Digital Signal processing†, IEEE Press. 2. Oran E. Brigham, â€Å"FFT and its Applications†, Prentice Hall, 1988. 3. www. mathworks. com amp; www. ti. com 4. www. wikipedia. com 5. Digital signal processing, salivanan