PEAK MODULATION CONTROL IN DIGITAL TRANSMISSION SYSTEMS Digital Overshoot: Causes, Prevention, and Cures David L. Hershberger, Greg J. Ogonowski, and Robert A. Orban PORTIONS COPYRIGHT © 1998 CONTINENTAL ELECTRONICS, INC. GRASS VALLEY, CA USA PORTIONS COPYRIGHT © 1998 MODULATION INDEX DIAMOND BAR, CA USA PORTIONS COPYRIGHT © 1998 ORBAN, INC. SAN LEANDRO, CA USA ALL RIGHTS RESERVED. NON-COMMERCIAL DISTRIBUTION ENCOURAGED. ABSTRACT Overmodulation can occur in both analog and digi- tal systems. Just because a system is digital does What are the causes of overshoot, and how is over- not make it immune to overshoot. Like analog sys- shoot different in digital systems? This tutorial an- tems, a digital audio system also contains subsys- swers these questions and will help you avoid the tems that can cause overshoot, like clippers and pitfalls in building a digital audio processing and filters. transmission system for FM broadcasting. DIAGNOSING OVERSHOOT INTRODUCTION How does one find overshoot in a system, and how Overshoot means at least two different things. In a can we tell the difference between overshoot and device designed to control instantaneous levels sloppy peak control? (like a peak limiter) overshoot occurs when the in- stantaneous output amplitude exceeds the nominal Overshoot detection methods are the same for both threshold of peak limiting. In a linear system de- analog and digital systems because both technolo- signed to pass a peak-controlled signal, overshoot gies ultimately produce the same kind of output: an occurs when the instantaneous system output am- analog FM signal. plitude exceeds the instantaneous input amplitude, assuming that the system has unity gain when One way to detect the problem is to use a good measured with a sinewave. Overshoots are unde- modulation monitor and a variable persistence os- sirable because they produce either (a) overmodu- cilloscope. Observe the composite output of the lation or (b) loss of loudness (if the modulation level modulation monitor with the oscilloscope. What you is decreased to avoid overmodulation). should see are frequent peaks reaching 100% modulation and nothing beyond. In a system with 150 overshoot, you will see occasional, low energy peaks extending typically 1-3 dB above the 100% 100 modulation level. The variable persistence feature 50 of the oscilloscope will help you to see these low energy peaks. 0 -50 If you do not have a variable persistence oscillo- MODULATION [%] scope, you can still detect overshoot by using the -100 peak flasher on the modulation monitor. In a system -150 with no overshoot, the threshold between no 0 0.002 0.004 0.006 0.008 0.01 flashes and continuous flashes should be abrupt. TIME For example, setting the flasher at 100% should Fig. 1 SINGLE OVERSHOOT WITH COMPLEX INPUT result in almost continuous flashes, and setting it at 103-105% should result in no flashes. However, if Figure 1 shows an overshoot. Most of the signal the range between continuous flashes and no peaks in this figure are accurately limited to 100%, flashes is 10% or more, then you have an over- but one isolated peak extends to about 125% shoot problem (which could possibly be in the modulation. modulation monitor). OVERSHOOT MECHANISMS elements in the processor can also control the overshoot. The requirements for peak control and spectrum control tend to conflict in audio processors, which is If the sharp-cutoff filter is now allowed to be mini- why sophisticated non-linear filters are required to mum-phase, it will exhibit a sharp peak in group achieve highest performance. Applying a peak- delay around its cutoff frequency. (A minimum controlled signal to a linear filter usually causes the phase filter is one that cannot have any less phase filter to overshoot and ring because of two mecha- shift for a given amplitude response.) Because the nisms: spectrum truncation and time dispersion. filter is no longer phase-linear, it will not only re- move the higher harmonics required to minimize 2 peak levels, but will also change the time relation- 1.5 ship between the lower harmonics and the funda- mental. They become unequally delayed, which 1 causes the shape of the waveform to change. This 0.5 time dispersion will further increase the peak level. 0 Figure 3 shows the amplitude and group delay -0.5 functions of a minimum phase, elliptic function filter. MAGNITUDE -1 Figure 4 shows the squarewave response of this filter. -1.5 -2 0 0.001 0.002 0.003 0.004 0.005 0.006 When a square wave is filtered by a linear-phase TIME lowpass filter, ringing will appear symmetrically on Fig. 2 FOURIER EXPANSION of a SQUAREWAVE the leading and trailing edge of the waveform. If the filter is minimum-phase, the overshoot will appear Peak limiters tend to produce waveforms with flat on the trailing edge and will be about twice as large. tops, like square waves. One can build a square In the first case, the overshoot and ringing are wave by summing its Fourier components together caused by spectrum truncation, which eliminates with correct amplitude and phase. Figure 2 shows harmonics necessary to minimize the peak level of the first three Fourier components of a squarewave. the wave at all times. In the second case, the over- Analysis shows that the fundamental of the square shoot and ringing are caused by spectrum trunca- wave is approximately 2.1dB higher than the ampli- tion and by distortion of the time relationship be- tude of the square wave itself. As each harmonic is tween the remaining Fourier components in the added in turn to the fundamental, its phase is such wave. Figure 5 compares the squarewave re- that the peak amplitude of the waveform decreases. sponses of a minimum phase filter and a linear Simultaneously, the R.M.S. value increases be- phase filter. cause of the addition of the power in each har- monic. This is the main theoretical reason why A worst-case scenario is shifting the phase of each simple clipping is such a powerful tool for improving harmonic in the square wave by 90 degrees the peak-to-average ratio of broadcast audio: clip- (performing a Hilbert Transform). The peak ping adds to the audio waveform spectral compo- amplitude of the square wave’s edges becomes, nents whose phase and amplitude are precisely theoretically, infinite. In the real world of band- correct to minimize the waveform’s peak level while limited audio waveforms this extreme overshoot simultaneously increasing the power in the wave- cannot occur. Nevertheless, it is perfectly possible form. to have phase shifts introduce up to 6dB of overshoot to peak-limited audio. This is equivalent Even a perfectly phase-linear low-pass filter will to 200% modulation! cause overshoot. If a square wave (or clipped waveform) is applied to a low-pass filter whose time delay is constant at all frequencies, the filter will remove the higher harmonics that reduce the peak level, increasing the peak level and thus the peak- to-average ratio. There is no sharp-cutoff linear low- pass filter that is overshoot-free; overshoot-free spectral control to FCC or ITU-R standards must be achieved by filters that are embedded within the processing so that the non-linear peak-controlling 2 10 0.0008 0.5 0 0.4 J 0.0007 J -10 0.3 J J 0.0006 J J -20 J J J J J J J 0.2 J J J J J J J J -30 J J J J 0.0005 J J JJ J J J J J J 0.1 J J J J J J J J J J -40 JJ J J J J J JJ J J J J J J 0.0004 0 J J J J J J JJ -50 J J J J J -0.1 J J J J J JJ J -60 0.0003 J J AMPLITUDE J J J J J J -0.2 J J J J J J J MAGNITUDE [dB] -70 J J J J 0.0002 J J GROUP DELAY [Sec] J J J -0.3 J -80 J J J J 0.0001 -90 -0.4 -100 0 -0.5 0 5000 10000 15000 20000 0 0.0005 0.001 0.0015 0.002 0.0025 FREQUENCY [Hz] TIME Fig. 3 MINIMUM PHASE LPF FILTER RESPONSE Fig. 6 SAMPLING MISSES THE PEAKS 0.5 2 0.4 1.5 0.3 1 0.2 J JJ J JJJJ J JJ J J JJ JJJ 0.5 0.1 J J J J J J J J J J J J J J J J J J J J J J J J J J J JJ J J J 0 J J J 0 JJ J J J J J J J J JJJJJ J J J J J J J J J J -0.1 J J J J J J J J J J J JJJJ J AMPLITUDE J -0.5 J J J J J J J J MAGNITUDE -0.2 J JJJ J J J J J J JJJJ -1 J JJ -0.3 JJ J J J J J -1.5 -0.4 JJ -2 -0.5 0 0.001 0.002 0.003 0.004 0.005 0.006 1.42E-5 3.54E-4 TIME TIME Fig. 4 MINIMUM PHASE LPF SQUAREWAVE RESPONSE Fig. 7 SAMPLING MISSES THE PEAKS—Expanded View 2 Overshoots and signal peaks may not coincide with 1.5 the sampling instants. In other words, the digital 1 samples may straddle the continuous waveform’s 0.5 peaks—the analog peak can “fall between the 0 samples.” Consequently, the continuous signal’s amplitude may exceed the amplitude of the digital -0.5 MAGNITUDE samples.