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Lucas Cordiviola
2022-04-12 23:32:40 -03:00
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@@ -30,50 +30,50 @@ original version by: Nikos Drakos, CBLU, University of Leeds
<BODY >
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<H1><A NAME="SECTION001130000000000000000"></A>
<A NAME="sect7.network"></A>
<H1><A ID="SECTION001130000000000000000"></A>
<A ID="sect7.network"></A>
<BR>
Delay networks
</H1>
<P>
<DIV ALIGN="CENTER"><A NAME="fig07.03"></A><A NAME="7867"></A>
<DIV ALIGN="CENTER"><A ID="fig07.03"></A><A ID="7867"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.3:</STRONG>
A delay network. Here we add the incoming signal to a delayed
@@ -95,7 +95,7 @@ successive moments in time, then time shifting the signal by <IMG
SRC="img28.png"
ALT="$d$"> samples
corresponds to a
<A NAME="7870"></A>
<A ID="7870"></A>
<I>delay</I>
of <IMG
WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
@@ -105,7 +105,7 @@ of <IMG
SRC="img36.png"
ALT="$R$"> is the sample rate.
Figure <A HREF="#fig07.03">7.3</A> shows one example of a
<A NAME="7873"></A><I>linear delay network</I>:
<A ID="7873"></A><I>linear delay network</I>:
an assembly of delay units, possibly with amplitude
scaling operations, combined using addition and subtraction. The output
is a linear function of the input, in the sense that adding two signals at the
@@ -117,13 +117,13 @@ so that the gains and delay times do not change with time.
<P>
In general there are two ways of thinking about delay networks. We can think
in the
<A NAME="7875"></A><I>time domain</I>,
<A ID="7875"></A><I>time domain</I>,
in which we draw waveforms as functions of time (or of the index <IMG
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img75.png"
ALT="$n$">), and
consider delays as time shifts. Alternatively we may think in the
<A NAME="7877"></A><I>frequency domain</I>,
<A ID="7877"></A><I>frequency domain</I>,
in which we dose the input with a complex sinusoid (so that its output is a
sinusoid at the same frequency) and report the amplitude and/or phase change
wrought by the network, as a function of the frequency. We'll now look at the
@@ -131,7 +131,7 @@ delay network of Figure <A HREF="#fig07.03">7.3</A> in each of the two ways in t
<P>
<DIV ALIGN="CENTER"><A NAME="fig07.04"></A><A NAME="8340"></A>
<DIV ALIGN="CENTER"><A ID="fig07.04"></A><A ID="8340"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.4:</STRONG>
The time domain view of the delay network of Figure <A HREF="#fig07.03">7.3</A>.
@@ -155,7 +155,7 @@ delayed copy of itself.
<P>
A frequently used test function is an
<A NAME="7886"></A><I>impulse</I>,
<A ID="7886"></A><I>impulse</I>,
which is a pulse lasting only one sample. The utility of this is that, if we
know the output of the network for an impulse, we can find the output for
any other digital audio signal--because any signal <IMG
@@ -271,7 +271,7 @@ combined additively.
<P>
<DIV ALIGN="CENTER"><A NAME="fig07.05"></A><A NAME="8344"></A>
<DIV ALIGN="CENTER"><A ID="fig07.05"></A><A ID="8344"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.5:</STRONG>
Analysis, in the complex plane, of the frequency-domain behavior of
@@ -483,19 +483,19 @@ The quantity <IMG
WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img30.png"
ALT="$\vert H\vert$"> is called the
<A NAME="7917"></A><I>gain</I>
<A ID="7917"></A><I>gain</I>
of the delay network at the angular frequency <IMG
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img27.png"
ALT="$\omega $">, and is graphed in
Figure <A HREF="#fig07.06">7.6</A>. The frequency-dependent gain of a delay network (that
is, the gain as a function of frequency) is called the network's
<A NAME="7920"></A><I>frequency response</I>.
<A ID="7920"></A><I>frequency response</I>.
<P>
Since the network has greater gain at some frequencies
than at others, it may be considered as a
<A NAME="7922"></A><I>filter</I>
<A ID="7922"></A><I>filter</I>
that can be used to separate certain components of a sound from others.
Because of the shape of this particular gain expression as a function of
<IMG
@@ -503,11 +503,11 @@ Because of the shape of this particular gain expression as a function of
SRC="img27.png"
ALT="$\omega $">, this kind of delay network
is called a (non-recirculating)
<A NAME="7924"></A><I>comb filter</I>.
<A ID="7924"></A><I>comb filter</I>.
<P>
<DIV ALIGN="CENTER"><A NAME="fig07.06"></A><A NAME="8359"></A>
<DIV ALIGN="CENTER"><A ID="fig07.06"></A><A ID="8359"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.6:</STRONG>
Gain of the delay network of Figure <A HREF="#fig07.03">7.3</A>, shown as a function
@@ -596,36 +596,36 @@ over the entire range of possible delay times.
<P>
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<ADDRESS>