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Basic Goertzel (EE Tip #120)

The basic Goertzel transform, which has been around since 1958, was derived from the discrete Fourier transform (DFT). It’s an extremely efficient method of detecting a single frequency component in a block of input data. Figure 1 depicts the signal flow for the basic Goertzel algorithm as each sample is processed.

Figure 1:  Figure 1—The signal flow of the basic Goertzel transform produces an output (y0) for each sample processed. The output is a combination of the current ADC sample added to the product of the previous output (y1) multiplied by a constant minus the previous output (y2). After a block of samples has been processed, the sum of the squares of y1 and y2 are computed to determine the relative amplitude of a particular frequency.

Figure 1: The signal flow of the basic Goertzel transform produces an output (y0) for each sample processed. The output is a combination of the current ADC sample added to the product of the previous output (y1) multiplied by a constant minus the previous output (y2). After a block of samples has been processed, the sum of the squares of y1 and y2 are computed to determine the relative amplitude of a particular frequency.

Figure 1 may be written as Equation 1. y0 is the current processed output. x0 is the current ADC sample. y1 is the previously processed output, and y2 is the next previously processed output. m is the frequency domain bin number. N is the sample block size. Input samples are processed on a sample-by-sample basis. Processing continues over a block of input data of the length N. After a block of data has been processed, magnitude and phase information for the signal of interest is contained in the y1 and y2 variables.

Equations 1 - 4

Equations 1 – 4

One of the advantages of the Goertzel transform is that the magnitude (and phase) computation needs to be performed only at the end of a block of data. After a block of data has passed through Equation 1, the relative magnitude may be determined by computing the sum of the squares of the real and imaginary components.

The real component (RC) is described as Equation 2. The imaginary component is calculated with Equation 3. The squared magnitude is computed with Equation 4.

For tone detection only, phase information may be ignored. It’s sufficient to compute the square of the magnitude to determine the relative power of a signal.—Eric Kiser, “Digital Decoding Simplified,” Circuit Cellar 182, 2005. The issue is available in the CC Webshop.

 

 

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Issue 282: EQ Answers

PROBLEM 1
Construct an electrical circuit to find the values of Xa, Xb, and Xc in this system of equations:

21Xa – 10Xb – 10Xc = 1
–10Xa + 22Xb – 10Xc = –2
–10Xa – 10Xb + 20Xc = 10

Your circuit should include only the following elements:

one 1-Ω resistor
one 2-Ω resistor
three 10-Ω resistors
three ideal constant voltage sources
three ideal ammeters

The circuit should be designed so that each ammeter displays one of the values Xa, Xb, or Xc. Given that the Xa, Xb, and Xc values represent currents, what kind of circuit analysis yields equations in this form?

ANSWER 1
You get equations in this form when you do mesh analysis of a circuit. Each equation represents the sum of the voltages around one loop in the mesh.

PROBLEM 2
What do the coefficients on the left side of the equations represent? What about the constants on the right side?

ANSWER 2
The coefficients on the left side of each equation represent resistances. Resistance multiplied by current (the unknown Xa, Xb, and Xc values) yields voltage.
The “bare” numbers on the right side of each equation represent voltages directly (i.e., independent voltage sources).

PROBLEM 3
What is the numerical solution for the equations?

ANSWER 3
To solve the equations directly, start by solving the third equation for Xc and substituting it into the other two equations:

Xc = 1/2 Xa + 1/2 Xb + 1/2

21Xa – 10Xb – 5Xa – 5Xb – 5 = 1
–10Xa + 22Xb – 5Xa – 5Xb – 5 = –2

16Xa – 15Xb = 6
–15Xa + 17Xb = 3

Solve for Xa by multiplying the first equation by 17 and the second equation by 15 and then adding them:

272Xa – 255Xb = 102
–225Xa + 255Xb = 45

47Xa = 147 → Xa = 147/47

Solve for Xb by multiplying the first equation by 15 and the second equation by 16 and then adding them:

240Xa – 225Xb = 90
–240Xa + 272Xb = 48

47Xb = 138 → Xb = 138/47

Finally, substitute those two results into the equation for Xc:

Xc = 147/94 + 138/94 + 47/94 = 332/94 = 166/47

PROBLEM 4
Finally, what is the actual circuit? Draw a diagram of the circuit and indicate the required value of each voltage source.

ANSWER 4
The circuit is a mesh comprising three loops, each with a voltage source. The common elements of the three loops are the three 10-Ω resistors, connected in a Y configuration (see the figure below).

cc281_eq_fig1The values of the voltage sources in each loop are given directly by the equations, as shown. To verify the numeric solution calculated previously, you can calculate all of the node voltages around the outer loop, plus the voltage at the center of the Y, and ensure they’re self-consistent.

We’ll start by naming Va as ground, or 0 V:

Vb = Va + 2 V = 2 V

Vc = Vb + 2 Ω × Xb = 2V + 2 Ω × 138/47 A = 370/47 V = 7.87234 V

Vd = Vc + 1 Ω × Xa = 370/47 V + 1 Ω × 147/47A = 517/47 V = 11.000 V

Ve = Vd – 1 V = 11.000 V – 1.000 V = 10.000 V

Va = Ve – 10 V = 0 V

which is where we started.

The center node, Vf, should be at the average of the three voltages Va, Vc, and Ve:

0 V + 370/47 V + 10 V/3 = 840/141 V = 5.95745 V

We should also be able to get this value by calculating the voltage drops across each of the three 10-Ω resistors:

Va + (Xc – Xb) × 10 Ω = 0 V + (166 – 138)/47A × 10 Ω = 280/47 V = 5.95745 V

Vc + (Xb – Xa) × 10 Ω = 370/47V + (138-147)/47A × 10 Ω = 280/47 V = 5.95745 V

Ve + (Xa – Xc) × 10 Ω = 10 V + (147-166)/47 A × 10 Ω = 280/47 V = 5.95745 V

Electrical Engineering Crossword (Issue 283)

The answers to Circuit Cellar’s February electronics engineering crossword puzzle are now available.

283-crossword-key

Across

2. LITZWIRE—Separately insulated strands woven together [two words]
4. LINKFIELD—First in a message buffer’s line [two words]
6. PETAFLOPS—Measures a processor’s floating point unit performance
8. ANION—negatively charged atom
9. LISP—Used to manipulate mathematical logic
11. STATCOULOMB—i.e., franklin (Fr)
12. AMBISONICS—Typically requires a soundfield microphone
15. BROUTER—This device can send data between networks and it can forward data to individual systems in a network
16. TRINITRON—This CRT technology was originally introduced the 1960s
17. OXIDE—The “O” in CMOS
18. DETENT—Used to prevent or stop something from spinning

Down

1. ELECTRICSUSCEPTIBILITY—XE [two words]
3. RADECHON—A barrier-grid storage tube
5. COLPITTS—This oscillator uses two-terminal electrical components to create a specific oscillation frequency
7. BEAGLEBOARD—TI’s open-source SBC
10. PICONET—A network that is created using a wireless Bluetooth connection
13. BETATRON—Designed to accelerate electrons
14. TEBIBYTE—More than 1,000,000,000,000 bytes

Circuit Protection (EE TIP #116)

Circuit protection is necessary to ensure that a circuit will work reliably for 10 years and beyond. Input power supplies are susceptible to spikes from various sources including lightning, high-power machinery (e.g., generators and motors), or interference from outside sources (e.g., microwaves). The figure below shows one way to provide circuit protection. If a voltage spike is applied at VIN, the metal oxide varistor (MOV) will act at 18 V and the positive temperature coefficient (PTC) fuse will limit the current drawn. A transorb (transient voltage suppressor), which can be thought of as an ultrafast silicon zener diode, can be used in place of the MOV. Also, a capacitor in parallel with the MOV will soak up fast transient spikes—an electrolytic capacitor for low-frequency transient voltages and a small-value ceramic capacitor for high frequency transient voltages.

Figure 1Editor’s Note: This EE Tip was written by Fergus Dixon of Sydney, Australia. Dixon, who has written two articles and an essay for Circuit Cellar, runs Electronic System Design, a website set up to promote easy to use and inexpensive development kits. Click here to read his essay “The Future of Open-Source Hardware for Medical Devices.”