RF knowledge that digital engineers need to master

With the need for mass data transmission and storage, more and more digital bus data rates have reached Gbit/s or above, such as HDMI data rate of 3.4Gb/s, USB3.0 data rate of 5Gb/s, SATA data rate of 6Gb/s, PCIE3.0 data rate of 8Gb/s, and more and more 10Gb/s or 25Gb/s are used for signal transmission in communications. The data rates of these digital signals have reached or even exceeded the frequency bands of radio frequency or microwaves that we traditionally call. In the process of transmission, real digital signals also increasingly show the characteristics of microwave circuits.

When analyzing these high-speed signals, traditional time-domain analysis methods face problems such as insufficient accuracy and lack of analysis methods, while frequency-domain analysis methods in the RF and microwave fields are very mature and complete. Therefore, more and more RF or microwave analysis methods are being used for the analysis and measurement of high-speed digital signals.

2. Bandwidth of digital signals

The primary reason for analyzing digital signals is that the real high-speed digital signals are far from the ideal 0/1 level in textbooks. There will definitely be some (even serious) distortion and deformation in the transmission process of real digital signals.

Figure 1. Difference between ideal and real digital signals.

To study digital signals, we must first obtain the real digital signal waveform, which involves the issue of the measuring instrument used. The best tool for observing the waveform of an electrical signal is an oscilloscope. When the signal rate is relatively high, the bandwidth of the oscilloscope generally required is also higher. If the bandwidth of the oscilloscope used is not enough, the high-frequency components in the signal will be filtered out, and the observed digital signal will also be distorted. Many digital engineers are accustomed to using harmonics to estimate the signal bandwidth, but this method is not very accurate.

For an ideal square wave signal, its rising edge is infinitely steep. From the frequency domain, it is composed of an infinite number of odd harmonics. Therefore, an ideal square wave can be considered as the superposition of an infinite number of odd sine harmonics.

However, for real digital signals, their rising edges are not infinitely steep, so the energy of their higher harmonics will be limited. For example, the following figure shows the spectrum of 50Mhz and 250MHz clock signals generated by the same clock source. We can see that although the output clock frequencies are different, the main spectrum energy of the signal is concentrated within 5GHz, which does not necessarily mean that the spectrum distribution of 250MHz is 5 times larger than that of 50MHz.

Figure 2. Spectra of clock signals with different frequencies generated by the same signal source

For real data signals, the spectrum is more complicated. For example, the envelope of the spectrum of the pseudo-random sequence (PRBS) code stream is a Sinc function. The figure below shows the spectrum of 800Mbps and 2.5Gbps PRBS signals generated by the same transmitter. We can see that although the output data rates are different, the main spectrum energy of the signal is concentrated within 4GHz, and it is not necessarily that the high-frequency energy of the 2.5Gbps signal is much higher than that of the 800Mbps signal.

Figure 3. Spectra of digital signals at different rates generated by the same signal source.

A spectrum analyzer is the most accurate tool for analyzing the frequency distribution of signal energy, so digital engineers can use a spectrum analyzer to analyze the spectrum distribution of the digital signal under test. When there is no spectrum analyzer available, we usually estimate the spectrum energy of the measured signal based on the rise time of the digital signal:

Maximum signal frequency content = 0.4/fastest rise or fall time (20 - 80%)

Or

Maximum signal frequency content = 0.5/fastest rise or fall time (10 - 90%)

3. The influence of transmission lines on digital signals

From previous research, we know that the spectrum of digital signals is widely distributed, and the range of its highest frequency component mainly depends on the rise time of the signal rather than just the data rate. When such a high-bandwidth digital signal is transmitted, the first challenge it faces is the influence of the transmission channel.

The bandwidth of real transmission channels such as PCB, cables, backplanes, connectors, etc. is limited, which will weaken or completely filter out the high-frequency components in the original signal. The loss of high-frequency components will manifest itself in the waveform as slowed edges, overshoot or oscillation of the signal, etc.

In addition, according to Faraday's law, the changing signal jump will generate eddy currents in the conductor to offset the change in current. The faster the rate of change of the current (equivalent to the shorter the rise or fall time of the signal for digital signals), the stronger the eddy current in the conductor. When the data rate reaches about 1Gb/s or more, the current of the signal in the conductor and the induced current are basically completely offset, and the net current is only limited to flow on the surface of the conductor. This is the skin effect. The skin effect will increase the loss and change the circuit impedance. The change in impedance will change the phase relationship of the harmonics of the signal, thereby causing signal distortion.

In addition, the FR-4 medium most commonly used to manufacture circuit boards is woven from glass fibers, which has poor uniformity and symmetry. At the same time, the dielectric constant of FR-4 material is also related to the signal frequency, so the transmission speed of different frequency components in the signal is also different. The difference in transmission speed will further change the phase relationship of each harmonic component in the signal, thereby further deteriorating the signal.

Therefore, when high-speed digital signals are transmitted on PCB, the high-frequency components of the signals will be weakened due to loss, and different frequency components will be transmitted at different speeds and superimposed on the receiving end. At the same time, some energy will be reflected multiple times at impedance discontinuities such as vias, connectors, or places where the line width changes. The combination of these effects will seriously change the shape of the waveform. It is a great challenge to analyze such a complex problem.

It is worth noting that many factors such as signal amplitude attenuation, changes in rise/fall time, and changes in transmission delay are related to frequency components, and different frequency components are affected differently. For digital signals, their frequency components are related to the digital symbols transmitted in the signal (for example, the frequency components represented by the code stream of 0101 and the code stream of 0011 are different), so different digital code streams are affected differently during transmission, which is inter-symbol interference ISI (ISI).

Figure 4. High-speed digital signal with severe intersymbol interference.

In order to analyze such a complex transmission channel, we can study its impact on the signal through the impulse response of the transmission channel. The impulse response of the circuit can be obtained by transmitting a narrow pulse. The ideal narrow pulse should be a narrow pulse with infinitely narrow width and very high amplitude. When this narrow pulse is transmitted along the transmission line, the pulse will be widened, and the shape after widening is related to the response of the line. Mathematically speaking, we can convolve the impulse response of the channel with the input signal to obtain the waveform of the signal after transmission through the channel. The impulse response can also be obtained through the step response of the channel. Since the differential of the step response is the impulse response, the two are equivalent.

It seems that we have found a solution to the problem. However, in reality, ideal narrow pulses or infinitely steep step signals do not exist. They are not only difficult to generate, but also difficult to control in accuracy. Therefore, in actual testing, sine waves are more often used to test to obtain frequency domain responses, and the time domain responses are obtained through the corresponding physical layer test system software. Compared with other signals, sine waves are easier to generate, and their frequency and amplitude accuracy are easier to control. Vector network analyzer VNA (vector network analyzer) can accurately measure the reflection and transmission characteristics of the transmission channel for different frequencies in a frequency range of up to tens of GHz by sweeping sine waves, with a dynamic range of more than 100dB. Therefore, modern high-speed transmission channel analysis mainly uses vector network analyzers for measurement.

The reflection and transmission characteristics of the system under test for sine waves of different frequencies can be expressed by S-parameters, which describe the transmission and reflection characteristics of the device under test for sine waves of different frequencies. If we can obtain the reflection and transmission characteristics of the transmission channel for sine waves of different frequencies, theoretically we can predict the impact of the real digital signal after passing through this transmission channel, because the real digital signal can be considered to be composed of many sine waves of different frequencies in the frequency domain.

For a single-ended transmission line, it contains 4 S parameters: S11, S22, S21, and S12. S11 and S22 reflect the reflection characteristics of port 1 and port 2 for sine waves of different frequencies, respectively. S21 reflects the transmission characteristics of sine waves of different frequencies from port 1 to port 2, and S12 reflects the transmission characteristics of sine waves of different frequencies from port 2 to port 1. For differential transmission lines, since there are 4 ports in total, its S parameters are more complicated, with a total of 16. Generally, a vector network analyzer with 4 or more ports is used to measure differential transmission lines to obtain their S parameters.

Figure 5. S-parameter model of a differential transmission line.

If the 16 S parameters of the measured differential line are obtained, many important characteristics of the differential line have been obtained. For example, the SDD21 parameter reflects the insertion loss characteristics of the differential line, and the SDD11 parameter reflects its return loss characteristics.

We can further obtain more information by performing an inverse FFT transformation on these S parameters. For example, the SDD11 parameter transformation can obtain the time domain reflection waveform (TDR: Time Domain Reflection), which can reflect the impedance change on the measured transmission line. We can also perform an inverse FFT transformation on the SDD21 result of the transmission line to obtain its impulse response, thereby predicting the waveform or eye diagram of digital signals with different data rates after passing through this pair of differential lines. This is very useful information for digital design engineers.

Figure 6. Channel insertion loss measured by a vector network analyzer and the signal eye diagram analyzed

Using a vector network analyzer (VNA) to measure the transmission channel of digital signals, on the one hand, borrows the analysis method of radio frequency microwaves, and can obtain very accurate characteristics of the transmission channel within the frequency range of tens of GHz; on the other hand, by performing some simple time domain transformation on the measurement results, we can analyze the impedance changes on the channel and the impact on the real signal transmission, so as to help digital engineers judge the quality of backplanes, cables, connectors, PCBs, etc. in the early stage, without having to wait until the signal has problems before rushing to deal with them.
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4. Signal Processing Technology

Since the impact of ISI on the transmission channel can be predicted by accurately measuring the characteristics of the transmission channel in advance, it is possible to correct it. Pre-emphasis at the transmitter and equalization circuit at the receiver are the two most common methods to compensate for the impact of channel transmission. The most obvious impact of the transmission channel is its low-pass characteristic, which will greatly attenuate high-frequency signals. For a square wave signal, its higher harmonics have a great impact on the shape of the signal. If all the higher harmonics are attenuated, the square wave will look like a sine wave.

Pre-emphasis is a method of compensating the high-frequency components of the transmitted signal in advance at the transmitting end. This method increases the amplitude of the first bit (transition bit) after the signal transition edge (pre-emphasis). For example, for a sequence of 00111, the amplitude of the first 1 in the sequence after pre-emphasis will be larger than the amplitude of the second and third 1s. Since the transition bit represents the high-frequency component in the signal, this method helps to increase the high-frequency component in the transmitted signal. In actual implementation, sometimes the amplitude of the transition bit is not increased, but the amplitude of the non-transition bit is reduced accordingly. This method is sometimes called de-emphasis.

Figure 7. Effect of pre-emphasis on signal.

When the signal rate is further increased or the transmission distance is longer, the pre-emphasis technology at the transmitter alone can no longer fully compensate for the loss caused by the transmission channel. At this time, equalization technology needs to be used at the receiver to improve the signal quality to ensure correct 0/1 judgment. There are three common signal equalization technologies: CTLE (continuous time linear equalizer), FFE (feed forward equalization) and DFE (decision feedback equalizer).

CTLE provides a bandpass filter at the receiving end, which can amplify the main high-frequency components in the signal, which is similar to the effect of the pre-emphasis technology at the transmitting end. FFE corrects the amplitude according to the weighted value of the voltage amplitude of adjacent bits. The weighting coefficient of each adjacent bit is directly related to the impulse response of the channel. CTLE and FFE are both linear equalization technologies, while DFE is a nonlinear equalization technology. DFE technology corrects the decision threshold of the current bit by the decision level of the adjacent bit. A well-designed DFE can effectively compensate for the impact of ISI on the signal. However, the premise for DFE to work correctly is that the 0/1 level of the adjacent bit is correctly judged, so there are certain requirements for the signal-to-noise ratio of the signal. Generally, CTLE or FFE is used to open the signal eye diagram first, and then DFE is used for further optimization.

Figure 8. Equalization improves the signal eye diagram.

5. Signal Jitter Analysis

Jitter reflects the time deviation of a digital signal from its ideal position. The bit period of high-frequency digital signals is very short, usually in the hundreds of ps or even tens of ps. Even a small jitter will cause the level of the signal sampling position to change, so high-frequency digital signals have strict requirements for jitter.

Figure 9. Definition of jitter.

The actual signal is very complex, and may contain both random jitter components (RJ) and deterministic jitter components (DJ) of different frequencies. Deterministic jitter may be caused by inter-symbol interference or some periodic interference, while a large part of random jitter comes from the noise on the signal. The figure below shows a noisy digital signal and its judgment threshold. Generally, we judge the state of the digital signal exceeding the threshold as "1" and the state below the threshold as "0". Since the rising edge of the signal is not infinitely steep, the vertical amplitude noise will cause the signal to change left and right when it passes the threshold point. This is the reason why the signal jitter is caused by noise.

Figure 10. Random jitter caused by amplitude noise.

To analyze signal jitter, the most commonly used tool is a broadband oscilloscope with corresponding jitter analysis software. The jitter analysis software in the oscilloscope can easily decompose the size and various components of the jitter, but due to the limitations of noise and measurement methods, it is difficult for the oscilloscope to accurately measure sub-ps jitter. Now many high-speed chips require the clock jitter to be less than 1ps or even lower. This requires the use of other measurement methods such as phase noise measurement methods.

We know that jitter is a deviation in time, and it can also be understood as a change in the clock phase, which is phase noise. For the clock signal, we observe the spectrum distribution of its fundamental wave. The spectrum of the fundamental wave of an ideal clock signal should be a very narrow spectrum line, but in fact, due to the existence of phase noise, its spectrum line is a relatively wide envelope. The narrower the envelope, the smaller the phase noise (jitter), and the closer the signal is to the ideal signal. The figure below is a spectrum of a real clock signal. The fundamental wave of the signal is at 2.5GHz. We observe the spectrum of 10MHz bandwidth near 2.5GHz. We can see that first of all, the spectrum of the signal is not a very narrow spectrum line, and its spectrum line is widened (the influence of random noise). Secondly, there are some interferences of specific frequencies superimposed on it (the influence of deterministic jitter).

Figure 11. Spectrum near the clock carrier signal as seen on a spectrum analyzer.

In order to more conveniently observe low-frequency interference, the signal carrier frequency is usually used as the starting point in phase noise measurement, and the horizontal axis is displayed logarithmically. The horizontal axis reflects the distance from the signal carrier frequency, and the vertical axis reflects the ratio of the energy of the corresponding frequency point to the signal carrier energy. The smaller this ratio is, the smaller the energy of other frequency components except the carrier, and the purer the signal. The instrument used to accurately measure the phase noise of the clock signal is the signal source analyzer. The signal source analyzer has a special circuit inside. Through multiple correlation processing of two independent local oscillators, the phase noise of its own local oscillator can be suppressed to a very low level, so that accurate phase noise measurement can be performed.

Figure 12. Phase noise of a clock signal measured by signal source analysis.

For many clocks generated by crystal oscillators, the main component of their jitter is random jitter. If we integrate the phase noise energy of different frequency components in the phase noise test results, we can get random jitter. By measuring the phase noise with a signal source analyzer and integrating the energy within a certain bandwidth, we can get accurate random jitter measurement results. The minimum jitter that a signal source analyzer can measure can be down to the fs level.

VI. Conclusion

From the above discussion, it can be seen that as an engineer who designs or tests high-speed digital circuits, there are many limitations to using traditional time domain methods to study signals and transmission channels. However, if you have some knowledge of RF and microwave, digital engineers can use spectrum analyzers to analyze signal spectra to understand the frequency distribution of signals and bandwidth requirements; they can use vector network analyzers to analyze the S parameters of transmission channels to understand the impedance changes of channels, reflections and losses at different frequencies, and predict the impact on signals; they can understand the compensation effects of pre-emphasis, equalization and other technologies on high-frequency losses; and they can use signal source analyzers to perform more accurate clock jitter measurements. These mature analysis methods and measurement methods in the field of RF and microwave can provide more useful information for digital engineering to have a deeper understanding of its high-speed signals, and further expand the analysis capabilities of digital engineers for high-speed signals.