Spatial Frequency Audio Frequency Responses

A good sound system is supposed to perform sound reproduction between the frequencies of 20 and 20,000 Hz (20 kHz). A frequency of 20 Hz is a rumble you almost feel instead of hear; 20,000 Hz is a fingernails-on-blackboard squeak. Outside these limits, it was once thought, humans do not hear well enough to make reproducing tones in recorded music worthwhile. More recently, audiologists have discovered that part of the information required to make what we hear seem real is contained somewhat outside these limits. Nevertheless, high fidelity sound reproduction is conventionally contained between 20 Hz and 20 kHz. Tones lower in frequency than 20 Hz are called infrasound, and tones higher in frequency than 20 kHz are called ultrasound. The analogy to "infrared" and "ultraviolet" is obvious. Think about what the 20-20,000 Hz limits mean. As long as a tone has a frequency between these limits, the audio electronics will reproduce it without too much loss or unusual gain. Once frequencies go into the infrasound or ultrasound range, the electronics are allowed to fail badly and the sound is degraded or vanishes. In other words, even if a higher frequency tone were put in the front end of the audio electronics, little or nothing is reproduced at the speaker end.

A similar effect can be seen in telescopes, but in optical systems the most interesting effect does not take place in the domain of optical frequencies, or "colors." It occurs for angle.

When a telescope is pointed at a white picket fence not too far distant, the gaps can be viewed crisply against a darker background. If, instead, you tape a small piece of paper with alternating white and dark bars on the same fence and look at it through the telescope, you are less likely to see the bars reproduced well. At some fine scale determined by moving the telescope closer or farther away, you will see the bars dissolve into a gray blur. The instrument is no longer reproducing the reality that may be verified by moving closer to the object. You know that the telescope is looking at bars, but it is no longer transmitting them as distinct stripes. Here, the effect is very close to the inability of audio electronics to reproduce too high a frequency, only in this case a portion of the object cycled between dark and light and back to dark again as different angles were considered. What is the analogy to frequency here? In the case of an audio tone, the standard units of frequency are cycles/second or hertz (named after 19th-century physicist Heinrich Hertz). In the case of astronomical images, the units are cycles/arcsecond. The resemblance to frequency is so unmistakable that the quantity used to describe the transmission of detail in an image is called spatial frequency. Spatial frequency has units of cycles/angle, but it is sometimes stated in cycles/distance in the focal plane-something like "200 line-pairs/mm" or just "lines/mm." It usually appears without the more precise "in the focal plane" trailer.

One might wonder if a low frequency cutoff exists in optical systems analogous to the 20 Hz cutoff of audio systems. You might think that none exists because dark and white bars are easier to see at low frequency. An effective cutoff is given by the limitations of the field, however. Once fewer than one bar shows in the lowest magnification eyepiece, the bars cannot really be said to resolve. Put another way, the illumination variation probably cannot be distinguished from the inherent vignetting of the system. A visual telescope of focal ratio f/8 has a maximum spatial frequency of 226 lines/mm at the focal plane. If that focal plane has a width of 25 mm inside the eyepiece, the lowest spatial frequency is certainly greater than 1 line/50 mm, or 0.02 lines/mm. The ratio of these two numbers is 11,300. Thus, the telescope has at least the 3 orders-of-magnitude bandwidth of the typical 20-20,000 Hz audio system.

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