The post PID controllers for beginners appeared first on The Tech Goggler.

]]>The output signal m(t) and the transfer function G(s) of the regulator, placed e(0)=0, are respectively equal to:

As seen, the ideal PID controller is an improper system, because when the pulsation ω tends to infinity and when ω=0. To limit the gain, in high and low frequency, the real PID regulator represented in the figure is used, whose transfer function is:

The designer must calculate the value of the K_{P}, K_{I} e K_{D} coefficients so that they meet the specifications of the frequency response (phase and gain margin, bandwidth, etc.) and those of the time response (full error, settling time, delay time, etc.).

Among the many methods available, this method is quite widespread in the industrial field. It consists in obtaining the optimal values of the K_{P}, K_{I} e K_{D} parameters by acting on special knobs of the regulator calibrated at the factory.

The phases of optimal regulation are:

- set K
_{P}=0, K_{I}=0 e K_{D}=0 and the adjustment ring is closed; - the value of the K
_{P}parameter is gradually increased, after excluding the derivative and integral actions, until the system reaches the limit of stability; - the value of K
_{P}=K_{Pmax}is measured, for which the response of the system to the unit step is an oscillation of constant amplitude; - the pulsation values are measured ω
_{c}and the T_{c}period of the persistent oscillation; - adjust the other knobs so that the parameters K
_{P}, K_{I}and K_{D}take the values shown in the following table.

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]]>The post Linear position transducer appeared first on The Tech Goggler.

]]>Analog position transducers transform a linear or angular movement into an electrical physical quantity. They are devices used for measurements of position, pressure, force, etc.

The linear position transducer consists of a resistance enclosed in a metal container in which a rod flows. Outside the container, there are 3 leads: start, center and end.

The resistance of the transducer is measured between the start and center terminal. It can vary between 0 Ω, rod in position L=0, and the value Rmax, rod in position L=Lmax.

Features | Values | Units of measurement |
---|---|---|

Face value | 2 | kΩ |

Rod stroke | 0-50 | mm |

max current intensity | 10 | mA |

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]]>The post The TIG (Tungsten Inert Gas) welding method appeared first on The Tech Goggler.

]]>The entire welding zone including the arc, the electrode, the molten base metal and the filler metal is immersed in an inert environment, consisting of argon gas, or helium or mixtures of the two: this fact prevents the oxidation of the welded joints by the atmosphere and thus allows the welding of particularly reactive materials.

The equipment required for TIG welding is schematically illustrated in Figure 1. The tungsten electrode is contained in an electrode gun and emerges from it in the central part of a nozzle from which the protective gas escapes. The gun is connected to the electric machine (operating on both direct and alternating current) with a sheath containing the electric cable connecting the electrode, the protective gas supply pipe coming from a cylinder through a special pressure regulator and economizer (to limit its consumption), the pipes for the forced circulation of the cooling water of the gun itself.

The electric arc is produced by the passage of electric current through the ionized gas: it follows that the arc is crossed by positive ions that move towards the negative pole and by electrons that move in the opposite direction. If you operate in direct polarity (as in most cases), that is, with the electrode at the negative pole, the ion bombardment of the electrode increases its temperature and contributes to the emission of electrons with particularly low voltages and arc currents, which represents an advantage in the welding of thin thicknesses.

The best method to ignite the arc is the one that involves the use of a pilot spark, caused by a high voltage and frequency, which shells between the elèttrodo and the piece when they are at close range. The spark causes the ionization of the protective gas and the ignition of the arc. This high voltage is automatically eliminated a few seconds after the arc is switched on if it is welded in direct current, while it is kept superimposed on the welding current in the case of alternating current, so as to stabilize the arc, as described below. At the end of the welding, an automatic system progressively reduces the welding current until the arc is turned off, while the protective gas is delivered for a few seconds until the joint is completely solidified and the electrode is sufficiently cooled.

The constituent material of the electrode can be 99.5% pure tungsten, or tungsten with 1-2% thorium, or tungsten with 0.15-0.4% zirconium. The former are the cheapest and are used in less critical welds, due to the low current density and sensitivity to contamination by molten metal splashes. The latter has greater emissivity at the same temperature, so they allow to weld with lower currents, have a greater duration and greater resistance to contamination. They also form a more stable and more easily triggered arc. Thirds have intermediate characteristics between the first two, but are generally preferable by welding in alternating current.

It has been said above that electrical machines for TIG welding have the possibility of delivering both direct and alternating current: the choice of the type of current depends essentially on the material to be welded.

- Direct current in direct polarity: the flow of electrons is directed towards the base metal, the thermal balance of the arc involves about 70% of the heat developed on the base metal. The arc is well concentrated, and penetration reaches its maximum value (within the framework of the TIG method). Since the electrode collects about 30% of the heat generated by the arc, it is not subjected to high-temperature increases, so under these conditions, it is possible to reach values of current density bearable by the pan electrode at about 60 A / mm2.
- Direct current in reverse polarity: the flow of electrons is directed towards the electrode and therefore the thermal balance of the arc is in favor of the electrode, which collects about 70% of the heat emitted: this involves a high overheating that forces the use of large diameter electrodes to dissipate this heat and places limitations on the maximum density of current obtainable (about 5 A / mm2) and obviously on penetration. In addition, 1’emission of electrons by the base metal creates a little concentrated arc. The only advantage in the use of this type of power supply would be the destruction of the layer of oxides that are formed in the case of welding magnesium and aluminum alloys thanks to the bombardment of positive ions. However, the minuses are predominant with respect to this advantage, so this power supply is practically not used.
- Alternate current: this case represents an intermediate situation between the previous two. In fact, the thermal balance of the arc involves 50% heat on the electrode and about 50% on the base metal. The penetration and concentration of the arc are intermediate between the two previous cases and so is the maximum current density that can be supported by the electrode (about 30 A/mm2). The destruction of the oxide layer occurs only in the semicycle where the electrode is positive. However, the use of alternating current creates arc stability problems. Three facts tungsten and base metal have different emissive power and this causes an unbalanced alternate current between one semicycle and another, with serious consequences on the stability of the arc. This fact is remedied by superimposing on the welding current a current with low intensity but with high voltage and frequency, in practice the same used to generate the pilot spark for the ignition of the arc.

TIG welding is a very effective method that allows obtaining high-quality welds on almost all metal materials (with the exception of those with low melting temperatures such as tin, lead and zinc). In particular, it is very suitable for aluminum alloys (in which the possible formation of A1203, highly refractory would cause problems in the joint), magnesium and reactive materials such as titanium and zirconium.

It is particularly suitable for thin thicknesses since the strong emissivity of the tungsten electrode allows to maintain stable arcs with low currents (up to about 15-20 A). The use in the field of medium and large thicknesses is certainly possible, but the TIG method is not economically convenient compared to other methods (MIG and submerged arc) characterized by greater penetrations and hourly depositions.

The TIG process is an expensive welding method, both for the cost of machines and equipment, for that of the tungsten electrode, and, above all, for that of the gases used (argon or helium). It is generally reserved for the finest materials and/or the thinnest thicknesses. By welding with the TIG method, it is possible that the tungsten of the electrode contaminates the base metal, forming hard and fragile inclusions.

Photo by Benjamin Wedemeyer

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]]>The post Bluetooth can be used to track people, according to research appeared first on The Tech Goggler.

]]>Mobile devices, such as phones, smartwatches, and fitness trackers, constantly transmit Bluetooth beacon signals at a rate of about 500 beacons per minute.

These beacons enable features such as Apple’s “Find My” lost device tracking service, COVID-19 tracing apps, and connectivity between smartphones and other devices such as wireless earphones. Prior research has revealed the existence of wireless fingerprinting in WiFi and other wireless technologies. The UC San Diego team discovered that this type of tracking can also be accomplished using Bluetooth.

“This is important because in today’s world Bluetooth poses a more significant threat as it is a frequent and constant wireless signal emitted from all our personal mobile devices,”

said Nishant Bhaskar, a Ph.D. student in the UC San Diego Department of Computer Science and Engineering and one of the paper’s lead authors.

The team of researchers from the Departments of Computer Science and Engineering and Electrical and Computer Engineering presented their findings on May 24, 2022, at the IEEE Security & Privacy conference in Oakland, California.

All wireless devices have minor manufacturing flaws in the hardware that make each device unique. These fingerprints are an unintended consequence of the manufacturing process. These flaws in Bluetooth hardware cause distinct distortions that can be used as a fingerprint to track a specific device. In the case of Bluetooth, this would allow an attacker to get around anti-tracking measures such as constantly changing the address a mobile device uses to connect to Internet networks.

It is difficult to track individual Bluetooth devices. Prior WiFi fingerprinting techniques rely on the fact that WiFi signals contain a long known sequence called the preamble. However, Bluetooth beacon signal preambles are extremely brief. “Because of the short duration, previous techniques are ineffective for Bluetooth tracking,” said Hadi Givehchian, a UC San Diego computer science Ph.D. student and lead author of the paper.

Instead, the researchers devised a new method that does not rely on the preamble and instead examines the entire Bluetooth signal. They created an algorithm for estimating two different values found in Bluetooth signals. These values vary depending on Bluetooth hardware flaws, providing researchers with the device’s unique fingerprint.

Several real-world experiments were conducted by the researchers to evaluate their tracking method. In the first experiment, they discovered that 40% of 162 mobile devices seen in public places, such as coffee shops, were uniquely identifiable. They then expanded the experiment and observed 647 mobile devices in a public hallway over the course of two days. The researchers discovered that 47 percent of these devices had unique fingerprints. Finally, the researchers demonstrated an actual tracking attack by fingerprinting and following a study volunteer’s mobile device as it walked in and out of their house.

Although their discovery is concerning, the researchers also discovered a number of difficulties that an attacker will face in practice. Temperature changes, for example, can affect the Bluetooth fingerprint. Certain devices also send Bluetooth signals with varying degrees of power, which influences how far these devices can be tracked. Researchers also point out that their method requires a high level of expertise from an attacker, so it is unlikely to be a widespread threat to the public today.

Despite the difficulties, the researchers discovered that Bluetooth tracking is likely feasible for a wide range of devices. It also does not necessitate sophisticated equipment: the attack can be carried out with equipment costing less than $200.

So, what’s the best way to address the issue? Bluetooth hardware would need to be rebuilt and updated from the ground up. However, the researchers feel that there are other, more straightforward options available. The team is presently developing a method to conceal Bluetooth fingerprints in Bluetooth device firmware via digital signal processing.

Researchers are now looking at whether their technology may be used on other sorts of gadgets. “Today, every kind of communication is wireless, and every type of communication is in danger,” said Dinesh Bharadia, a professor at the UC San Diego Department of Electrical and Computer Engineering and one of the paper’s senior authors. “We’re focusing on hardware-based safeguards against possible threats.”

Researchers discovered that just turning off Bluetooth does not always prevent all phones from broadcasting Bluetooth beacons. When turning off Bluetooth from the control center on the home screen of some Apple products, for example, beacons are still sent.

“As far as we know,” Bhaskar added, “the only thing that would certainly deactivate Bluetooth beacons is turning off your phone.” Even while researchers can track specific devices, they are unable to gather any information about the devices’ owners, according to the researchers.

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]]>The post Microsoft is fighting the war for recruitment appeared first on The Tech Goggler.

]]>However, admitting this is a business faux pas for both companies and job seekers. Microsoft Corp isn’t wasting any time in making a bold move to lure top talent: come work with us, and you’ll be earning $163,000 as a 22-year-old fresh out of college.

Being so open about remuneration could be troubling for a corporate leader. Doesn’t this mean you’ll attract people who are only interested in the money and don’t care about the company or the job?

This is untrue, according to an economic study. Financial incentives can motivate employees to be more engaged and loyal, not in spite of the money, but because of it.

Humans have a narrative bias as a result of the cognitive revolution. This means that we tell ourselves stories to make sense of our surroundings. One such tale is about money. Money, beyond its monetary value, helps us to imagine the tale we want our lives to be. It may seem self-evident, yet we all want to make meaning, and elite talent is no exception. The link between intelligence and creativity is added to this instinct. Clever talent may be more naturally creative in their stories, fantasies, and ambitions, and they’ll desire to exercise those muscles.

Microsoft not only provides them with a playground in which to do so, but it’s also likely to be their first exposure to this type of employer compensation, given that it’s aimed at recent grads. Given what we know about the anchoring effect, Microsoft is laying the groundwork for a future of devoted, knowledgeable employees who are enthralled from the start. A formula for success.

Our survival instinct is our most basic instinct. We examine our environs for immediate and future risks in any undertaking. In our daily lives, this could mean being unable to afford housing, healthcare, or family assistance. We are less cerebral, imaginative, and productive in this survival phase. Cortisol levels are higher in our bodies. We are more stressed and agitated. And doing so for lengthy periods of time causes us to lose our appetites, sleep patterns, and mental wellness. Your firm will suffer if your employees feel this way. Not only will their productivity plummet, but they’ll also abandon your company in search of better opportunities. This can create instability inside your organization and will be more costly in the long run. What is the solution? Simply put, greater remuneration.

Human societies have always been founded on the basis of negotiation. ‘Tit for tat’ cultures developed informal accountability between strangers within society through agreements. This accountability thrived because of a combination of human fears of social isolation and being labeled as a disobedient member of the community. What does this imply for your company?

It means that your staff is in the reputation management industry at the end of the day. You fulfilled your part of the contract by providing generous recompense, and now it’s their turn. They’ll begin to feel more responsible for their task, take more initiative to complete it, and are more likely to align with your company’s current status quo – lest they earn a bad reputation. What’s the end result? Employees who are engaged, productive, and loyal.

Photo by Blogging Guide

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]]>The post Analog temperature transducers: example of a circuit appeared first on The Tech Goggler.

]]>- high sensitivity;
- linear output;
- stability of operation over time

To discuss the operation of temperature transducers, we will proceed directly to illustrate some fundamental commercial transducers, without however going too far into the infinite variations that exist of each model.

The PT100 transducer, also called RTD thermoresistance, is a precision resistance whose RT value is a function of temperature T.

In general, the nominal value of the resistance of the device is 100 Ω. The R_{T}/T characteristic is linear and increasing:

where:

- R
_{T}is the value of the resistance to generic temeperature T; - R
_{0}is the nominal value of the temperature resistance at 0 °C; - α=3.85*10-3 °C
^{-1}is the average dimensional coefficient.

In practice, it will be necessary to perform a resistance/voltage conversion. For example, below is an R_{T}/V converter that uses a constant current generator. In this scheme, a voltage regulator is used LM317, capable of delivering a current I≥ 10 mA.

The constant current intensity is equal to:

and for I=10 mA we have:

For example, if T=20 °C, since from the datasheets we get R_{T}=107.79 Ω and imposing I=10 mA, the output voltage is V_{O}=1.077 V.

If the resistance thermometer is placed at a distance from the control system, the connecting wires introduce an additional resistance in the resistance/voltage conversion. In this regard, manufacturers produce three-wire and four-wire resistors suitable for eliminating the error due to the resistance of the connecting wires.

We realize a conditioning circuit for a PT100 temperature transducer able to provide an output voltage between 0 and 5 V when the temperature T varies in the range 0-50 °C.In the proposed shema the integrated INA111 is proposed.

Assuming that the output current from the operational is I=2 mA, each branch of the Wheatstone Bridge is crossed by a current of 1 mA such as not to self-heat the R_{T} resistance thermometers.

In the hypothesis that the CA3140 amplifier can be considered ideal, from the circuit represented in the figure we obtain:

For T=0 °C you have R=R_{T}(0 °C)=100 Ω and for T=50 °C you have R_{T}(50 °C)=119.25 Ω. So you have:

The G gain of the INA111 amplifier and the RG resistance are:

It is possible to perform the calibration of the circuit with this procedure:

- act on the trimmer R
_{l}to fix an I =2 mA; - replace the PT100 with two precision resistors of 100 Ω±0.1% to simulate the operating condition of the transducers for a temperature T =0 °C. In this condition one must have V
_{o}=0 V; - now replace the two previous precision resistors with two 119.25 Ω to simulate the operation of the transducers for a temperature of T = 50 ° C. In this condition one must have V
_{o}=5 V. In fact, at this temperature, one has R_{T}=100(1+αT)=100(1+3.85*10^{-3}*50)=119.25 Ω

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]]>The post How do you solve difference equations? appeared first on The Tech Goggler.

]]>If the latter is not available, it can be derived by iteratively solving the difference equation with an input u(k)=δ(k) and null inputs and outputs for k<0.

In the case where the initial conditions are not null, it is possible to obtain y(k), given the sequence u(k) and the N values of the output prior to the instant k = 0, with a procedure very similar to that used for the solution of linear differential equations with constant coefficients for continuous systems over time.

For the linearity of the system, you can consider the output y(k), for k≥0, as the sum of the forced response to the input signal starting from -M, and the free-evolution of the system, that is, u(k)=0 for every k and y(-1), y(-2), … y(-N) equal to the given initial values, namely:

The free evolution response can be calculated with:

This is called the **associated homogeneous equation**.

The general solution of the associated homogeneous equation is constructed from the roots of the **characteristic equation**:

If p is a simple root of the characteristic equation, it is easy to verify that the sequence satisfies the associated homogeneous equation.

If C(λ)=0 has N distinct roots, the general solution of the homogeneous equation is given by the linear combination of linearly independent sequences , that is:

The constants shall be calculated so that y(k) assumes the values given in k=-1….-N.

In the case that the root p has multiplicity r, it contributes to the general solution with the linear combination of r linearly independent sequences:

where j=0,1,2…r-1

For r=3, we get the sequences , , .

In the case of simple but complex roots, they will be in complex conjugate pairs, since the coefficients of C(λ) are real, and in the same way the coefficients c must also be complex conjugates.

It is a linear equation to the differences of order 1, in fact N = 1, M = 0.

The homogeneous equation results:

and the characteristic equation:

which has a single simple root a. The general solution then becomes:

If one wishes to evaluate the free evolution from the initial condition y(-1)=0.7, the coefficient c must be calculated so that:

which results:

We can also proceed through the calculation of the impulse response of the system. We can derive h(k) from the difference equation and it is:

where the factor is to make h(k) causal.

The total response y(k), starting from a given value of y(-1) and with generic input u(k), is given by:

where the second term represents the forced response of the system.

Consider now the latter with step input, i.e. :

When , diverges, and when it converges to the value .

For the response is monotonous increasing, while for the response is oscillatory, as shown in the following figures in which (b=1-a) has been placed.

We determine the h(k) samples needed to calculate y(k) for k=0,1,2,3,4,5. That is, let’s determine the impulse response:

[1]To evaluate the samples h(k) we place in input a pulse: u(0)=δ(0)=1 and solve iteratively:

Starting from:

we iteratively perform the calculation of the response and compare it with the previous solution, remembering [1].

As we can see, the results of the two methods coincide.

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]]>The post Discrete systems and difference equations appeared first on The Tech Goggler.

]]>The system is also causal in that the output at instant k depends only on the previous outputs and inputs.

In this article we will consider only discrete time-invariant systems in which, that is, f{•} does not explicitly depend on k and of the finite-dimensional type, for which the output at instant k depends on a finite number of previous inputs and outputs:

If we define n=max{N,M} as the order of the system, the equation is called difference equation of order n.

In the case of multivariable systems, y(k) and u(k) will be vectors of appropriate size, and the relation input-output will be described by a system of difference equations.

Similarly to continuous systems over time, much of the analysis and synthesis of discrete systems is carried out in the context of Time-Invariant Linear Discrete Systems, indicated by DLIT. For DLIT systems the representation becomes:

where the coefficients are real.

The output y(k) is a linear combination of the previous N outputs and the previous M inputs, in addition to the current u(k) input if .

The representation of a DLIT system with the recursive difference equation is also called the Autoregressive Moving Average (ARMA) model. It corresponds to the linear differential equation with constant coefficients for linear and continuous systems over time.

The determination of the output y(k) given the input sequence u(k) can be done by iteratively solving the difference equation of order n, starting from the initial conditions, that is, the M+N previous values of the input and output.

The difference equation of order n can then be associated with the following calculation procedure:

R and S are memory areas in which the previous M+N input and output values necessary for the calculation of y(k) are stored.

Discrete systems have become increasingly important as computer applications for real-time signal processing have increased. In a DDC control, the control algorithm developed by the µP can be described as a discrete system, very often of the DLIT type, since the synthesis of the regulator is conducted within linear systems (neglecting the quantizations).

However, there are many other applications in areas other than control, where you have to deal with discrete systems. An example is the field of digital signal processing, which is of great interest in the fields of telecommunications and data analysis (instrumentation). In general, an iterative calculation procedure that processes a sequence of input data can be represented as a discrete system and vice versa.

Consider, as an example, the algorithm for determining the square root of a real number U. As is known, the calculation can take place iteratively by solving this equation with Newton’s method:

which leads to the following algorithm:

This equation represents a discrete system of order 1, non-linear and time-invariant, subjected to a constant input U, with initial condition given by y(-1), which corresponds to an initial estimate of .

As k increases, the output tends to .

The fact that a calculation algorithm can be thought of as a discrete system **made it possible to apply system stability theorems to the problems of numerical convergence of iterative algorithms**.

Given the enormous interest in discrete systems in the field of digital signal processing (digital filters) appeared, in the late 70s, special microprocessors, based on a different architecture, which allow the efficient calculation of the difference equation of order n that describes a DLIT.

These devices, called DSPs (Digital Signal Processors) execute the following instruction very quickly (tens of nanoseconds):

**AR=AR+COEFF*VAR**

where AR is the accumulator register, COEFF a coefficient (a_{i},b_{j}) and VAR a variable (y(i), u(j)).

We now consider the h(k) impulse response of a DLIT system, i.e. the output sequence in response to an input signal equal to δ(k) from null initial conditions. Since the input sequence u(k) can be expressed by:

for the linearity of the system the output at instant k is given by the sum of the impulsive responses of the system in response to the various inputs u(i)δ(i), that is:

which represents the discrete convolution operation between the two signals u(k) and h(k).

For causal systems h(k)=0 for k<0, it turns out:

DLIT systems for which the impulse response is canceled for all k above a finite value, are called FIR (Finite Impulse Response); otherwise, they are called IIR (Infinite Impulse Response).

The forced response y_{f}(k), that is, starting from null initial conditions, to a causal input u(k) applied at the initial instant (k=0), becomes:

Given the invariance over time of the system, this expression is generally valid considering as the initial instant that of application of the input.

The calculation of the forced response of a DLIT system to a u(k) input can essentially take place in two ways:

- iteratively solving the equation to differences of order n;
- resorting to the discrete convolution of the input with the impulse response of the system.

We will see these methods in a future article.

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]]>The post Recursive filters (IIR) appeared first on The Tech Goggler.

]]>Translating into words the [1], we can say that the output of the filter is equal at any moment to a fraction α of the input at that instant, added to the output in the previous instant multiplied by the complement of α.

This mechanism is called recursive, by analogy to what happens in the computer field, since the output at instant n is a function of the output at instant n-1 which in turn is a function of the output at instant n-2, and so on.

From [1] we see that the α factor has the role of a time constant because it tells us how much the output in the current state depends on the output in the previous state and therefore we know an estimate of the speed with which the output itself varies.

In fact, for α = 1 the output depends only on the entrance at that moment (zero time constant), while for α = 0 the output is always the same as that in the previous instant and therefore there is no evolution, whatever the input (infinite time constant).

Rarely, however, will you use a filter with such a slow response as that of an RC-type analogic filter, so digital filters make sense from the second order onwards.

The description of a second-order recursive filter is provided by the expression:

[2]In [2], the output in instant n is a function of the input at the same instant and the output in the previous two moments. But the output in the two previous moments was a function of the input in those moments and of the outputs in the two moments even earlier. Taking the speech to the limit, it can be concluded that the output at a certain moment is somehow a function of the outputs in all the previous moments.

Reasoning in this way, we can say that **the filter has an infinite impulse response**, hence the name of IIR (Infinite Impulse Response).

The transfer function of a second-order IIR filter is obtained, similarly to what is seen for FIR filters, by applying the z transform to the finite differences equation and then looking for the ratio between output and input in the z domain.

Applying the z transform to [2], we get:

from which the transfer function will result:

In a future article we will generalize the transfer function for IIR filters and compare them with FIR filters.

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]]>The post The discrete transfer function of a FIR filter appeared first on The Tech Goggler.

]]>Starting from the discrete Fourier transform and applying simplification we can define:

This new form in which the discrete Fourier transform can be placed is called the **zeta transform**.

Without getting, for now, into the topic, (see this previous article), keep in mind that the most important effect of this transformation is that every elementary delay in time, equals in the domain of the transform to a division by z. For example:

and more generally:

Applying the zeta transform thus defined to the finite difference equation that defines a generic FIR filter over time, (see this previous artile), we get an algebraic equation that constitutes its definition in the complex domain of the variable z.

or:

At this point it is immediate to derive the filter transfer function:

that is, in a more compact form:

[1]The last two equations represent two equivalent ways of writing the discrete transfer function of a generic FIR filter of order N-1.

FIR filters are very important because you can always make sure that they have a linear phase shift with the frequency.

Although the phase of the transmitted signals is often neglected in monoaural telecommunications systems, such as the telephone, in the case of multiphonic systems it is of fundamental importance in the perception of the position of the sound.

Hence the need for certain applications to have linear phase shift filters with frequency so that the phase is not distorted. This is analogous to what happens in power lines when the Heaviside condition is valid since the physical means have the task of carrying the signal without distorting it.

In fact, if an x(t) signal is transmitted along a line, it is considered not distorting if the Heaviside conditions are respected, that is, if it is admitted that the signal itself can be recovered from the one received y(t) at less than a scale factor and a finite time delay, namely:

Considering the revealing property of the pulse, it is easy to verify that this is achieved by seeing the line as a filter whose response to the pulse is worth:

Therefore, to derive the Heaviside conditions valid in frequency, it is enough to transform the g(t) obtaining:

From this expression we can say that for a filter the Heaviside condition corresponds to the fact that its transfer function has constant amplitude and phase proportional to the frequency for all frequencies; this is completely equivalent to what applies to transmission lines.

For an FIR filter to introduce a linear phase shift, its coefficients must be symmetrical if the filter is low-pass and antisymmetrical if the filter is high-pass.

Without going into the demonstration, the filter transfer function [1] can be rewritten as:

[2]where, starting from [1] we placed (normalized frequency), the change of variable and then applied Euler’s formulas.

Now, from [2], we see that if the coefficients a_{k} are symmetric, i.e. a_{k}=a_{-k}, then in the summation the sinusoidal functions disappear and G(θ) will remain real and multiplied by a phase shift factor equal to which varies linearly with frequency.

If, on the other hand, the a_{k} coefficients are antisymmetric, i.e. a_{k}=-a_{-k}, the cosinusoidal functions will disappear from the summation. The phase shift will still be linear, due to the presence of the term although increased by one factor since the function is now purely imaginary. Also note that, in the latter case, it will become an odd function of the frequency and then the filter will be of the high-pass type.

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]]>