Introduction to Frequency Domain TIPL 4301 TI Precision Labs – ADCs Hello, and welcome to the TI Precision Lab covering frequency domain analysis for data converters. This video will introduce the concept of the frequency domain. It will also cover common AC specifications such as SNR and THD. Created by Art Kay, Luis Chioye Presented by Peggy Liska

Time Domain vs. Frequency Domain Fourier Spectrum for Vout 1Vpk @ 1kHz 1Vpk @ 1kHz 0.1Vpk @ 10kHz Sum of both signals 0.1Vpk @ 10kHz Here we introduce the concept of time domain and frequency domain. Time domain shows the amplitude of signals versus time. In the time domain example we are creating an unusual wave shape by adding two sine waves together. A 1Vpk sine wave at 1kHz and a 0.1Vpk waveform at 10kHz. We will look at this combined waveform in the frequency domain. In the frequency domain amplitude is on the vertical axis and frequency is on the horizontal axis. In the frequency domain a “Fourier Spectrum” plot always shows the amplitude of sinusoidal waveforms versus frequency. In this example, we are analyzing the signal Vout which is composed of two sinusoidal waveforms. Thus the frequency domain has two components at 1kHz and 10kHz with amplitudes of 1Vpk and 0.1Vpk respectively. The key point here is that the frequency domain only displays the amplitude for sinusoidal waveforms. So how do non-sinusoidal waveforms show up in the frequency domain? In the next slide we will see how all waveforms can be represented as a sum of sinusoidal waveforms. Time Domain Frequency Domain

Infinite Series for a Triangle Wave 𝑓 𝑡 = 8 𝜋 2 𝑘=0 ∞ (−1) 𝑘 sin⁡( 2𝑘+1 𝜔𝑡) 2𝑘+1 2 𝑓 𝑡 = 8 𝜋 2 sin 𝜔𝑡 − 1 9 sin 3𝜔𝑡 + 1 25 sin 5𝜔𝑡 −… Any arbitrary waveform can be decomposed into an infinite series of sinusoidal waveforms. This infinite series is called a fourier series. An example of a fourier series for a triangle waveform is shown here. The mathematical function shows that the series starts with sin(ωt) and continues with 1/9*sin(3ωt), 1/25*sin(5ωt) and so on. The first and largest frequency component, sin(ωt), is called the fundamental. Subsequent sinusoidal components such as sin(3ωt) and sin(5ωt) are called harmonics and they will diminish in amplitude as the series continues. Theoretically, to get a perfect representation of a triangle wave an infinite number of sine waves is required. However practically if you use the fundamental and several harmonics you will get a pretty good representation of the triangle wave. This example shows a fundamental with four harmonics in both the time and frequency domain. Notice in the time domain waveform the triangle waveform at the bottom which is the direct sum of the waveforms above. The frequency domain of the triangle waveform is generated using a mathematical transformation called a fourier transform. The fourier transform is a general way to transform any time domain signal into it’s frequency domain equivalent. Time Domain Frequency Domain

Dynamic Characteristics In this slide we introduce the transfer function for a data converter and its associated quantization error. In subsequent slides we will see the impact of quantization noise in frequency domain analysis. An example of a four bit transfer function is shown at the left. On the horizontal axis is the analog input signal and on the vertical axis is the digital equivalent code. The digital output is a rounded off version of the analog input. Sweeping the analog input voltage you can see that each digital output code has one unique analog input where the error is zero. Moving above or below this analog signal causes the error to increase or decrease to a maximum value of half an LSB bit width. This error due to the digital rounding is called quantization error. On the right hand side you can see a digitized sine wave versus the continuous analog input. At the bottom of the page you can see the quantized error for the sine wave. The quantization error can be thought of as a noise, because it is a error that is superimposed on the digitized sine wave. The total rms quantization noise can be determined by integrating the quantization error function. Let’s look at the frequency domain relationship for this waveform.

Signal to Noise Ratio (SNR) On the left hand side you can see the frequency domain representation for the digitized sine wave. The digitized signal applied is at 1kHz and the noise is the quantization noise from the “rounding” error on the waveform. Here we are considering the ideal data converter, where all the noise is from quantization noise. In practical cases, other noise sources will contribute to the noise floor. The total rms quantization noise can be theoretically predicted by integrating the time domain quantization error waveform. In the next slide we will see how the noise floor can be used to calculate the SNR for an ideal “N” bit data converter.

Signal to Noise Ratio (SNR) Measured Ratio: 𝑆𝑁𝑅( 𝑉 𝑉 )= 𝑉 𝑆 𝑉 𝑁 Measured dB: 𝑆𝑁𝑅(𝑑𝐵)=20∙𝑙𝑜𝑔 𝑉 𝑆 𝑉 𝑁 Ideal ADC SNR: 𝑆𝑁𝑅 𝑑𝐵 =6.02∙𝑁+1.76 Where N is the number of bits e.g. N = 10 for a 10 bit converter This slide shows the general equation for signal to noise ratio, or SNR. In a general sense the signal to noise ratio is a measurement of how clean or noise free a signal is. A high signal to noise ratio indicates that the signal is very large in comparison to the noise whereas a low signal to noise ratio indicates that the noise is high relative to the signal. For this ratio, both the signal and noise are measured in volts rms. Take 20 times the log of the ratio to convert it to decibels. The ideal signal to noise ratio in decibels can be calculated by taking 6.02*N + 1.76, where N is the number of bits. A 10 bit converter, for example, would have 6.02*10 + 1.76 or 61.96dB SNR. This relationship was derived by integrating the quantization noise an applying the signal to noise relationship. This is the relationship for an ideal converter where the only error source considered is quantization noise. No practical data converter will have a better signal to noise than what is given by this equation, as practical converters have other noise sources.

Ideal SNR calculation from Resolution According to the previous equation: 𝑆𝑁𝑅= 6.02𝑁+1.76 𝑑𝐵 This table gives the ideal SNR for multiple different ADC resolutions. The values for SNR were generated using the equation SNR = 6.02*N + 1.76dB. For example, a 16 bit converter’s ideal SNR is 98.08dB. A practical 16 bit converter may achieve SNR close to 98.08, but it will never exceed this. The equation and associated table are a good way to know how close a particular converter is to ideal.

Nonlinearity Besides SNR, Total Harmonic Distortion, or THD, is a common data converter ac specification. In order to understand THD it is important to understand nonlinearity. Nonlinearity is a measurement of how much a transfer function deviates from its ideal straight line. The transfer function shown on the left hand side of the slide shows an ideal linear transfer function and a nonlinear transfer function. The ideal transfer function follows a straight line in the form y=mx+b whereas the nonlinear will have higher order terms causing deviations from the line. The nonlinear example shown is exaggerated to make the nonlinearity easy to see. Notice how the nonlinear function tracks well for low input levels and deviates as the input increases. In short, the gain for higher input signals is larger than it should be. This has the effect of stretching out the top half cycle of the sine wave. This stretching of the top half cycle is called distortion and will create harmonics in the frequency spectrum.

Total Harmonic Distortion (THD), SINAD This slide shows the frequency spectrum for the digitized sine wave at the right. The harmonics are the result of the distortion on the top half cycle of the waveform. Harmonic distortion will always occur at integer multiples of the fundamental frequency. In this case the fundamental is at 1kHz and there are harmonics at 2kHz, 3kHz, 4kHz and so on. Sometimes it is useful to differentiate between even and odd harmonics, as different circuit nonidealities may generate one type of harmonic. Even harmonics are even multiples of the fundamental frequency and odd harmonics are odd multiples of the fundamental. In this example, 2kHz and 4kHz are even harmonics whereas 3kHz and 5kHz are odd harmonics. If the digitized signal perfectly tracked the input signal, there would not be any harmonics.

Total Harmonic Distortion (THD), THD+N, SINAD 𝑇𝐻𝐷(%)= 𝑖=2 10 𝑉 𝑖 2 𝑉 𝑆 2 ∙100 𝑇𝐻𝐷(𝑑𝐵)=20∙𝑙𝑜𝑔 𝑖=2 10 𝑉 𝑖 2 𝑉 𝑆 2 (𝑇𝐻𝐷+𝑁)(𝑑𝐵)=20∙𝑙𝑜𝑔 𝑖=2 10 𝑉 𝑖 2 + 𝑉 𝑁 2 𝑉 𝑆 2 𝑆𝐼𝑁𝐴𝐷(𝑑𝐵)=20∙𝑙𝑜𝑔 𝑉 𝑆 2 𝑖=2 10 𝑉 𝑖 2 + 𝑉 𝑁 2 The THD calculation is given here as a percentage as well as in decibels. The IEEE standard for ADC testing specifies that the second through tenth harmonics should be used in the THD calculations. THD is the square root of sum of the harmonic voltages squared divided by the rms signal voltage squared. This quantity is multiplied by 100 to convert to percentage or 20 times the log is taken to convert to decibels. THD+N is similar to THD except that it includes the total rms noise in the calculation. SINAD is short for signal to noise and distortion. Mathematically, SINAD is simply the reciprocal of the THD+N calculation. In decibels, taking the reciprocal will just change the sign of the number. Note that SINAD or THD+N will always be worse than either the THD or SNR because SINAD is really a combination of the two error sources.

Total Harmonic Distortion (THD), THD+N It is important to note that the harmonic component included in the THD calculation is omitted from the SNR calculation. The harmonic component is replaced with the average noise at the frequency where the harmonics occur. The idea is that distortion and noise are two separate error sources and we don’t want to include distortion in the noise calculation.

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