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<H1><A NAME="SECTION001100000000000000000"></A>
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<A NAME="chapter-delay"></A>
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<BR>
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Time shifts and delays
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</H1>
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<P>
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At 5:00 some afternoon, put on your favorite recording of the Ramones string
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quarter number 5. The next Saturday, play the same recording at 5:00:01, one
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second later in the day. The two playings ideally should sound the same.
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Shifting the whole thing one second (or, if you like, a few days
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and a second) has no physical effect on the sound.
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<P>
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But now suppose you played it at 5:00 and 5:00:01 on the same day (on two
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different playback systems, since the music lasts much longer than one second).
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Now the sound is much different. The difference, whatever it is,
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clearly resides in neither of the two individual sounds, but rather in the
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<A NAME="7751"></A><I>interference</I>
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between the two. This interference can be perceived in at least four different
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ways:
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<DL>
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<DT></DT>
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<DD>Canons: Combining two copies of a signal with a time shift sufficient
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for the signal to change appreciably, we might hear the two as separate musical
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streams, in effect comparing the signal to its earlier self. If the signal is
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a melody, the time shift might be comparable to the length of one or several
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notes.
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<P>
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</DD>
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<DT></DT>
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<DD>Echos: At time shifts between about 30 milliseconds and about a second,
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the later copy of the signal can sound like an echo of the earlier one. An echo
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may reduce the intelligibility of the signal (especially if it consists of
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speech), but usually won't change the overall "shape" of melodies or
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phrases.
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<P>
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</DD>
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<DT></DT>
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<DD>Filtering: At time shifts below about 30 milliseconds, the copies are
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too close together in time to be perceived separately, and the dominant effect
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is that some frequencies are enhanced and others suppressed. This changes the
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spectral envelope of the sound.
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<P>
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</DD>
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<DT></DT>
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<DD>Altered room quality: If the second copy is played more quietly than the
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first, and especially if we add many more delayed copies at reduced amplitudes,
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the result can mimic the echos that arise in a room or other acoustic space.
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<P>
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</DD>
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</DL>
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The sound of a given arrangement of delayed copies of a signal may
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combine two or more of these affects.
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<P>
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Mathematically, the effect of a time shift on a signal can be described as a
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phase change of each of the signal's sinusoidal components. The phase shift of
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each component is different depending on its frequency (as well as on the
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amount of time shift). In the rest of this chapter we will often consider
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superpositions of sinusoids at different phases. Heretofore we have been
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content to use real-valued sinusoids in our analyses, but in this and later
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chapters the formulas will become more complicated and we will need more
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powerful mathematical tools to manage them. In a preliminary
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section of this chapter we will develop the additional background needed.
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<P>
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<BR><HR>
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<!--Table of Child-Links-->
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<UL>
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<LI><A NAME="tex2html2096"
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HREF="node105.html">Complex numbers</A>
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<UL>
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<LI><A NAME="tex2html2097"
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HREF="node106.html">Complex sinusoids</A>
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</UL>
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<BR>
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<LI><A NAME="tex2html2098"
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HREF="node107.html">Time shifts and phase changes</A>
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<LI><A NAME="tex2html2099"
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HREF="node108.html">Delay networks</A>
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<LI><A NAME="tex2html2100"
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HREF="node109.html">Recirculating delay networks</A>
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<LI><A NAME="tex2html2101"
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HREF="node110.html">Power conservation and complex delay networks</A>
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<LI><A NAME="tex2html2102"
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HREF="node111.html">Artificial reverberation</A>
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<UL>
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<LI><A NAME="tex2html2103"
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HREF="node112.html">Controlling reverberators</A>
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</UL>
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<BR>
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<LI><A NAME="tex2html2104"
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HREF="node113.html">Variable and fractional shifts</A>
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<LI><A NAME="tex2html2105"
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HREF="node114.html">Fidelity of interpolating delay lines</A>
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<LI><A NAME="tex2html2106"
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HREF="node115.html">Pitch shifting</A>
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<LI><A NAME="tex2html2107"
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HREF="node116.html">Examples</A>
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<UL>
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<LI><A NAME="tex2html2108"
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HREF="node117.html">Fixed, noninterpolating delay line</A>
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<LI><A NAME="tex2html2109"
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HREF="node118.html">Recirculating comb filter</A>
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<LI><A NAME="tex2html2110"
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HREF="node119.html">Variable delay line</A>
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<LI><A NAME="tex2html2111"
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HREF="node120.html">Order of execution and lower limits on delay times</A>
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<LI><A NAME="tex2html2112"
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HREF="node121.html">Order of execution in non-recirculating delay lines</A>
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<LI><A NAME="tex2html2113"
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HREF="node122.html">Non-recirculating comb filter as octave doubler</A>
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<LI><A NAME="tex2html2114"
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HREF="node123.html">Time-varying complex comb filter: shakers</A>
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<LI><A NAME="tex2html2115"
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HREF="node124.html">Reverberator</A>
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<LI><A NAME="tex2html2116"
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HREF="node125.html">Pitch shifter</A>
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</UL>
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<BR>
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<LI><A NAME="tex2html2117"
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HREF="node126.html">Exercises</A>
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<ADDRESS>
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Miller Puckette
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2006-12-30
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