How To Find Bandwidth From Frequency

Measuring and Characterizing Nonlinear Radio-Frequency Systems

Wendy Van Moer , ... Kurt Barbé , in Microwave De-embedding, 2014

vi.5.ane Adjacent co-channel power ratio

Frequency bandwidth is very scarce and expensive nowadays. Thousands of telecommunication applications are used daily, and they all crave their ain slice of bandwidth. Sharing these express frequencies requires that each application does not "leak" into the frequency ring of the other. In practice, this "frequency leaking" is caused by the nonlinear behavior of components and systems. Hence, when designing RF systems it is very important to know the amount of in- and out-of-band nonlinear distortions that will be generated by the devices. The amount of interference, or ability, in the adjacent frequency aqueduct is represented past the Adjacent co-Channel Power Ratio (ACPR), which is divers as the ratio of the boilerplate ability in the next frequency channel to the average power in the transmitted frequency channel [24,25]. The ACPR describes the level of distortions generated by the nonlinear beliefs of RF components and is often used to characterize the linearity of a device. A high ACPR corresponds to a strong nonlinear device.

When the input indicate of the device under exam contains nonlinear distortions due to for case source-pull, the ACPR will effect in an over- or underestimation of the level of nonlinear distortions. In society to compensate for these nonlinear input distortions, a linearized transfer function is required that is able to describe the behavior of the DUT in-band too as out of the frequency band where the device operates.

In the previous section, we accept shown that the BLA tin non just be determined inside but also exterior the frequency band where the DUT operates. To obtain this out-of-ring BLA, the system is jointly excited by a big-indicate and an additional small-signal that lies outside the BLA band. At present, we will show that this additional small excitation signal allows compensating for the nonlinear distortions nowadays in the input signal and, hence, results in a correct source-pull free level of nonlinear output distortions.

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Physical Sensors: Acoustic Sensors

Osamu Saito , ... Yoji Okabe , in Reference Module in Biomedical Sciences, 2021

Phase-shifted FBG for highly sensitive acoustic measurement

The frequency bandwidth and sensitivity of sensors are the most of import parameters to guarantee the accuracy and efficiency of the acoustic measurement organization built using the abovementioned sensing methods. Minardo et al. (2005) determined that the detected waveform is mostly plain-featured when the grating length L (Fig. 12B) is comparable to the wavelength of the ultrasonic betoken. Hence, to precisely detect acoustic waves that take a curt wavelength, or high frequency, the grating length must be shortened. However, a short grating length makes the slope of the edge in the FBG spectrum gentle, which leads to decreased sensitivity.

Stage-shifted fiber Bragg grating (PSFBG) has been widely introduced in the field of fiber optic acoustic sensing to overcome the trade-off between the frequency bandwidth of sensing and the sensitivity (Deepa and Das, 2020; Wu et al., 2018). The PSFBG is manufactured by inserting a phase shift into the periodic perturbation of the refractive index, as illustrated in Fig. 15A . The spectra of the PSFBG tin can be calculated using a transfer matrix method (Erdogan, 1997). Fig. xvB shows the calculation outcome for the reflectivity as a part of wavelength after a phase shift is inserted into the FBG; the corresponding spectrum without the stage shift is shown in Fig. 13B. A comparison between Figs. 15B and thirteenB indicates that an additional narrow dip (zoomed-in view in Fig. 16) with a steep linear slope and no side lobes tin exist generated in the center of the PSFBG reflectivity. The steep slope of the dip induces a more sensitive audio-visual detection.

Fig. fifteen. PSFBG sensor. (A) Stage shift of grating. (B) Calculation result of the reflection spectrum of a PSFBG (n _eff  =   one.45, Λ   =   534.v   nm, L  =   1   cm, and $\overline{\delta n_{\mathrm{eff}}}=4\times {10}^{-\mathrm{iv}}$ ) as a function of wavelength.

Fig. 16. Zoomed-in view of PSFBG reflectivity near the Bragg wavelength. Parameters are the aforementioned as that of Fig. xiiiB.

Wu and Okabe (2012) verified the high sensitivity of the PSFBG sensor for ultrasonic measurement. They experimentally compared iii detection methods (Fig. 14A–C); the results are shown in curves (i), (2), and (iii) in Fig. 17. The SNR value obtained from these ability spectra indicated that the PSFBG-balanced sensing system had the best SNR. The sensitivity of the balanced PSFBG sensing arrangement was mostly comparable to that of the PZT sensors.

Fig. 17. Ability spectral densities of the ultrasonic responses obtained from iii different sensing systems. (1) Traditional FBG sensing organisation based on power intensity method. (ii) Border-filtering-based PSFBG sensing system. (3) Balanced PSFBG sensing organization.

Reprinted with permission from Wu Q and Okabe Y (2012) High-sensitivity ultrasonic phase-shifted fiber Bragg grating balanced sensing system. Optics Express 20: 28353–28362. © The Optical Society.

In addition to the high sensitivity, the PSFBG has a broader frequency response than the FBG because its response to ultrasonic waves is primarily determined, not past all regions of the gratings but the nearby region of the stage-shifted area. That is, the constructive gauge length of the PSFBG is very short. Thus, the PSFBG sensor tin detect ultrasonic waves with curt wavelengths, which leads a broad bandwidth in the frequency response. The comeback in the sensing bandwidth was evaluated by an experiment using the border filtering system (Fig. 14B) for both the PSFBG and a mutual FBG. In the measurement, a chirped ultrasonic wave was used as an input wave over a frequency range of 10   kHz to 1.five   MHz. The detected waveforms are shown in Fig. 18A and B . The high-frequency wave components after 180   μs were detected by the PSFBG; however, they were not detected past the normal FBG. Their corresponding Fourier transfer results overlap in Fig. 18C. The frequency spectrum results show that the PSFBG sensor successfully accomplished an authentic response to the ultrasonic input from 10   kHz to 1.5   MHz. By dissimilarity, betoken detection by the FBG was limited to a frequency less than 0.6   MHz.

Fig. eighteen. Results of ultrasonic measurement with FBG and PSFBG. Their grating lengths are five   mm. 4096 waves were averaged. (A) Waveform measured with PSFBG. (B) Waveform measured with FBG. (C) Comparing of Fourier spectrum between PSFBG and FBG.

Modified from our original figures in Yu F and Okabe Y (2019) Regenerated fiber Bragg grating sensing system for ultrasonic detection in a 900 °C environment. Periodical of Nondestructive Evaluation, Diagnostics and Prognostics of Engineering Systems 2: 011006.

Past utilizing the high sensitivity and the broadband response of the PSFBG, many advanced optical cobweb audio-visual sensing systems comparable to PZT sensors accept been successfully constructed in diverse fields, such equally acoustic emission in a composite material (Wu et al., 2015; Yu et al., 2016a,b), acousto-ultrasonic measurement (Guo and Yang, 2015), nonlinear ultrasonic measurement (Wu et al., 2019), underwater audio-visual measurement (Rosenthal et al., 2011), ultrasonic imaging (Guo et al., 2014), and laser ultrasonic visualization (Yu et al., 2020).

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RF Propagation, Antennas, and Regulatory Requirements

Shahin Farahani , in ZigBee Wireless Networks and Transceivers, 2008

5.5.v Outcome of Bespeak Spreading on Multipath Performance

If the betoken frequency bandwidth is smaller than the coherent bandwidth, the entire signal spectrum will experience like fading. This means that if a portion of this signal spectrum is in deep fade, it is probable that the unabridged signal spectrum will be in deep fade. For example, in a large outdoor area with filibuster spread of 250   ns, the coherence bandwidth (B_C) is 637   kHz.If there is no spreading and the signal bandwidth (B_South) is 250   KHz, then B_S < B_C and the entire 250   KHz spectrum of the signal experiences the same fading. If the receiver happens to be in a multipath null, the unabridged signal spectrum is in deep fading.

Signal spreading increases the signal bandwidth, and when the betoken bandwidth becomes sufficiently larger than the coherence bandwidth, it is possible that when a portion of the signal spectrum is in deep fade the rest of the signal will experience a different, and potentially better, fading environment. In our earlier outdoor example, if the signal bandwidth (B_{Due south}) is increased to 2   MHz using betoken spreading, the indicate bandwidth exceeds the coherent bandwidth and a portion of the signal spectrum is always exterior any possible multipath null. A signal with partly disturbed spectrum may even so have a chance of recovery by the receiver.

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Super-Resolution Fluorescence Microscopy

Alberto Diaspro , ... Colin J.R. Sheppard , in Comprehensive Nanoscience and Nanotechnology (Second Edition), 2019

4.01.5 Unlimited Increase in the Spatial Frequency Bandwidth

Further increase in the spatial frequency bandwidth is commonly based either on a near field result, as in the scanning about-field optical microscope (SNOM) or photon tunnelling microscope [50–52], or on a nonlinear optical interaction with the sample (as well occurring in the near field) such as switching, blinking or saturation.

In fact, the two-fold increased bandwidth of fluorescence microscopy (over a brightfield microscope) can be explained by the square law procedure of photon absorption. Similarly, the increased bandwidth of two-photon microscopy, either by two-photon excitation fluorescence or SHG, can likewise be explained in this style.

In a brightfield microscope, there is no nonlinear interaction with the sample, so it seems that, according to nowadays do at least, it is not possible to obtain an unlimited increment in spatial frequency bandwidth in stage or polarization contrast.

In full general superresolution has been explained using information theory, in terms of the information chapters of an optical arrangement [53–57]. According to this perspective, resolution is not an invariant of a system, and superresolution tin can exist achieved past trading off some other cistron such as a priori knowledge, field of view, or acquisition time. Thus the digital office of prototype reconstruction should be considered as an integral role of the imaging process, every bit in the case of computational imaging.

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LTE-M

Olof Liberg , ... Joachim Sachs , in Cellular Internet of Things, 2018

5.4.i.6 Faster Frequency Retuning

Due to the reduced radio frequency bandwidth of depression-toll LTE-Yard devices, guard periods for frequency retuning are needed, every bit described in Sections 5.2.4.1 and 5.two.5.1. In Release 13, the device creates a guard period of ii OFDM/SC-FDMA symbols. In Release 14, it is possible for the device to indicate that it tin can practice faster frequency retuning so that the guard period can be smaller than 2 symbols. The device can indicate that it needs ane symbol or fifty-fifty nada symbols—the latter value is mainly intended for ordinary LTE devices, which may have no need to do retuning to move betwixt dissimilar narrowbands when operating in CE way (considering ordinary LTE devices tin receive and transmit the full LTE system bandwidth rather than just a narrowband or wideband). Faster retuning ways somewhat less truncation of the transmitted signal and therefore somewhat meliorate link functioning.

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Applications to Audio-Video Systems

Kunimaro Tanaka , in Essentials of Error-Control Coding Techniques, 1990

8.2.1 Characteristics of the Audio Data

Sound-signal quality requires that frequency bandwidth be more than xx kHz and that dynamic range be more than 80 dB. This will decide the minimum sampling frequency and its minimum coding bit number. In improver, physical limitations of the recorder restrict the selection of the sampling frequency. When a video recorder is used to record the audio data, it is desirable that an integer number of samples are recorded in one horizontal scanning-line catamenia of video signal. The sampling frequency should also be selected keeping the data-transmission aqueduct in listen.

Consequently, the standard sampling frequency chosen in various standards are 48, 44.i, 44.059, 32 and 31.v kHz. The standard bit numbers are viii, 14, 16, and twenty bits. The transfer rate of the audio information, therefore, is 96 kbyte/sec when sampling frequency is 48 kHz and coding bit number is xvi bits. When they are 31.five kHz and 8 bits, the transfer rate is 31.v kbyte/sec (Gibson, 1982).

The correlation of a sample value betwixt next samples is high when data are digital audio. Therefore, the erroneous sample tin exist error-compensated by such methods as the first-order interpolation, in which the erroneous sample value is replaced by boilerplate value of the preceding correct sample value and that of the following correct sample.

An erroneous sample should be sandwiched by correct samples for better fault bounty in decoding. Therefore, even-number samples and odd-number samples should be separated on the recording media when it is recorded. This is usually called shuffling. When size of the error is smaller than shuffling length, erroneous samples are sandwiched by correct samples on mistake compensation.

Figure 8.ii shows diverse types of error bounty. (A) shows muting in which erroneous samples are held zero. (B) illustrates the zeroth-order extrapolation in which each erroneous sample is held to the value of its preceding right sample. (C) explains the first-gild interpolation in which each erroneous sample is held to the average value of its preceding correct sample and correct following sample. Implementation of averaging is simple when data is linearly coded. Suppose the preceding sample has a value of Sa and the post-obit sample has a value of Sb, and so (Sa + Sb) tin can be obtained with a full adder. (Sa + Sb)/2 tin can be obtained by shifting the output of the full adder by one bit. (D) corresponds to higher-order interpolation in which erroneous sample is substituted past the value calculated from adjacent samples. ◯ shows the correct sample, and × corresponds to the erroneous sample in the effigy. The erroneous samples are sandwiched past right samples considering shuffling has worked correctly. The dotted line shows the original audio bespeak, and the solid line does the line connecting the tiptop of the compensated sample value. In that location exists a smoothing filter later the digital to analog converter. Therefore, the obtained playback signal is closer to the original line than the solid line in the figure. Although noise induced by fault compensation becomes smaller going from (A) to (D) as tin can be seen from the figure, implementation gets more complicated (Tsuiki et al., 1977).

Fig. viii.ii. Diverse types of error compensation. (A) Muting; (B) the zeroth-guild extrapolation; (C) the first-order interpolation; (D) the college-order interpolation.

Figure 8.three illustrates the noise caused by inaccuracy of the error bounty (Ohnishi et al., 1978). The vertical centrality corresponds to the signal-to-racket ratio, whose dissonance is induced by fault compensation. The horizontal centrality responds to the audio frequency normalized by the sampling frequency. The figure shows that the first-order interpolation makes less noise than the zeroth order extrapolation and the noise level is low in the low-frequency region. Information technology means that the level of the noise, which is induced by the inaccuracy of error bounty, depends upon the type of musical musical instrument. For example, the audio of a double bass can be error compensated ameliorate than that of a piccolo. In addition, the noise is more audible in classical music than in pop music. The first-gild interpolation tin can practically compensate with almost all music in general.

Fig. viii.3. Noise acquired by inaccuracy of the error compensation. Signal frequency f _o is normalized past the sampling frequency f _s.

The click racket acquired by miscorrection is hated extremely. Therefore, information technology is non good strategy to use the full ability of the fault-correcting code for error correction and cause miscorrection when there are a large size of errors in decoding. Sometimes it is more than practical to spare the ability for error detection. There was a recorder whose errors were controlled past only error compensation in the early stage of digital sound when semiconductors were still very expensive (Iga et al., 1977). On the other paw, there is generation loss when errors are controlled only past bounty, and this loss kills one of the advantages of the digital audio recorder. Considering semiconductor prices take come down, very strong error-correcting codes are employed in the recorders developed since 1983.

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Information Acquisition and Signal Processing

Alan S. Morris , Reza Langari , in Measurement and Instrumentation (Second Edition), 2016

6.half dozen Analog Betoken Processing

As noted before, the concepts of frequency spectrum and bandwidth are essential in understanding the nature of sensor signals. These notions are not only key to the way sensor signals are interpreted, but also essential to the way in which sensor signal processing mechanisms are devised. For instance, to eliminate dissonance, which has a high-frequency spectrum, a low-laissez passer filter is used. At that place too instances such as in processing linear variable differential transducer (LVDT) signals that a loftier-pass filter is utilized. Finally there are instances where the desired spectrum is constrained inside a certain range in which case a band-laissez passer filter is used. Finally, in that location are cases where a specific undesirable frequency such every bit the threescore   Hz power line frequency must be eliminated in which instance a notch filter is used.

In the sequel, we will hash out the pattern of passive and active filters (and amplifiers). Passive filters, equally the proper noun implies, are made up of passive components such as resistors and capacitors. They are simple to pattern and implement. Nevertheless, they can excessively load the source circuit and thereby undermine the indicate processing process. In addition, passive filters practise non provide for a robust design equally their frequency response characteristics (their functional bandwidth) varies with the characteristics of the circuitry in which they are integrated.

Active circuits (filters and amplifiers), on the other hand, incorporate operational amplifiers or transistors, and are robust and more constructive every bit signal processing devices. They do require a power source, which may or may not be an issue depending on the circumstances in which they are deployed. Active circuits are likewise more than circuitous to design and implement and are more costly. Nevertheless in almost applied indicate processing applications, agile filters (every bit filters or as amplifier/filters) are preferred to passive ones for the reasons noted higher up.

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Communications Satellite Systems

Takashi Iida , Hiromitsu Wakana , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

II.D.ane.c CDMA

In CDMA, all signals occupy the same frequency bandwidth and are transmitted simultaneously in time, but the different signals are distinguished from 1 another at the receiver by the specific spreading codes or frequency hopping blueprint. Spread spectrum multiple access (SSMA), which is the most popular CDMA, is achieved by 2 techniques: direct sequence (DS) modulation and frequency hopping (FH) modulation. In DS, the modulated signal is multiplied by pseudorandom noise (PN) codes with a chip rate of R _c, which is much larger than an information bit rate of R _b. The resulting indicate has wider frequency bandwidth than the original modulated signal. At a receiving final, the received signal is despread by multiplying the same PN sequence. In FH, bandwidth spreading is accomplished past pseudorandom frequency hopping. The hopping pattern and hopping rate are determined by the PN code and code rate, respectively.

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SPECTRAL Assay, CLASSICAL METHODS

S. Braun , in Encyclopedia of Vibration, 2001

The Doubtfulness Principle

The doubtfulness principle describes the relationship between time elapsing and the frequency bandwidth of signals. The quantitative formulation depends somewhat on how the 'elapsing' and 'bandwidth' are defined, and more than ane possibility exists for such definitions. For any reasonable definition: (fourth dimension elapsing) × (frequency bandwidth) > C where C is in the range of 1.five–3, depending on the definition. Resolving within f ₂ − f ₁ thus necessitates a duration of:

[13] $T_t\ge \frac{2}{f_{\mathrm{two}}-f_{\mathrm{ane}}}$

The dubiety principle is so basic that its applied implications are sometimes overlooked. Thus, given a point length T _t , ii components separated by f ₂ − f _one = 1/T _t will non be separated by any bespeak-processing techniques (unless additional information is at hand). Another firsthand determination is that changes in the frequency domain separated by f ₂ − f ₁ < C/T _t are meaningless, still the possibility of calculating at sometimes closer arbitrary frequencies (see below).

For digital FFT-based computations, the frequency scale is in steps of:

$\mathrm{\Delta }f=1/\left(N\mathrm{\Delta }t\right)=\mathrm{one}/T_t$

and the dubiousness principle is 'automatically' satisfied. Dedicated instrumentation sometimes have a 'zoom mode' for spectral analysis, whereby simply a partial frequency range is zoomed in for analysis. High resolution is achieved if Northward frequency points are computed in this range. Information technology should be clear that this can but be achieved using an appropriate data elapsing, and very large T _t may be needed with such a mode of operation. Specifically, if in zoom mode Due north frequencies are resolved in the range f _two − f ₁, then:

$T_t=1/\left(\mathrm{\Delta }f\right)=\mathrm{North}/\left(f_2-f_1\right)$

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Multicarrier transmission in a frequency-selective aqueduct

S.K. Wilson , O.A. Dobre , in Academic Press Library in Mobile and Wireless Communications, 2016

ix.3.2.ane Windowing and frequency characteristics

One of the cardinal bug in any modulation technique is its frequency spectrum and bandwidth. OFDM is made of multiple subcarriers, each with an energy spectrum that decays at a rate that is inversely proportional to the frequency (Fig. 9.8).

Fig. 9.8. Spectrum of an OFDM symbol with 128 subcarriers, along with that of a single subcarrier.

Consider an OFDM signal with a cyclic prefix:

$\begin{array}{ll}\hfill x(t)& =\sum _{\mathrm{one\; thousand}=0}^{N-1}{c}_g{\text{e}}^{j2\pi \frac{\mathrm{1000}t}{NT}}\sqcap \left(\frac{t}{NT+\mathrm{\Delta }}\right),\hfill \\ \hfill X(f)& =(NT+\mathrm{\Delta })\sum _{k=0}^{\mathrm{North}-1}{c}_k\text{sinc}\left(\left(f-\frac{\mathrm{yard}}{NT}\right)(NT+\mathrm{\Delta })\right),\hfill \\ \hfill G(f)& =E[|\mathrm{10}(f){|}^2]\hfill \\ \hfill & ={(NT+\mathrm{\Delta })}^2\sum _{\mathrm{yard}=0}^{N-1}{\mathcal{E}}_c\text{sinc}{\left(\left(f-\frac{\mathrm{chiliad}}{NT}\right)(NT+\mathrm{\Delta })\right)}^{\mathrm{ii}}.\hfill \end{array}$

To determine how quickly the sum rolls off exterior the main frequency window, annotation that for f > 0, $\text{sinc}{\left(\left(f-\frac{k}{NT}\right)(NT+\mathrm{\Delta })\right)}^{\mathrm{ii}}\le \frac{1}{{\pi }^2{(\mathrm{Northward}T+\mathrm{\Delta })}^2{\left(f-\frac{k}{NT}\right)}^2}$ ; and approximating the sum by an integral, for f outside the bandwidth, $\left[0,\frac{1}{T}\right]$ ,

$\begin{array}{ll}\hfill \mathrm{Thousand}(f)& \approx {(\mathrm{North}T+\mathrm{\Delta })}^2{\mathcal{E}}_c{\int }_0^{\mathrm{Due\; north}-1}\frac{\mathrm{i}}{{\pi }^{\mathrm{ii}}{(NT+\mathrm{\Delta })}^2{\left(f-\frac{\mathrm{ten}}{NT}\right)}^{\mathrm{two}}}d\mathrm{ten}\hfill \\ \hfill & ={\mathcal{E}}_c\frac{\mathrm{Due\; north}T}{{\pi }^{\mathrm{two}}}\frac{1}{f\left(f-\frac{\mathrm{North}-\mathrm{i}}{\mathrm{Northward}T}\right)}.\hfill \end{array}$

The chief point hither is that the frequency will still roll off past a cistron of 1/f ² exterior the bandwidth, but the driblet from 1 to $\frac{1}{{(NT+\mathrm{\Delta })}^2}$ is decreased by the factor of NT in the numerator. That is, by summing the North sinc-squared terms, we are increasing the out-of-band spectrum past a factor of North.

The square pulse is responsible for the relatively high out-of-ring spectrum values. One way to reduce the out-of-ring spectrum, and hence narrow the effective bandwidth, is the use of windowing.

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