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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Steepness of a transfer function with frequency, particularly in electrical network analysis</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about roll-off in electrical network analysis. For the dumpster, see <a href="/wiki/Roll-off_(dumpster)" title="Roll-off (dumpster)">Roll-off (dumpster)</a>.</div> <p><b>Roll-off</b> is the steepness of a <a href="/wiki/Transfer_function" title="Transfer function">transfer function</a> with <a href="/wiki/Frequency" title="Frequency">frequency</a>, particularly in <a href="/wiki/Network_analysis_(electrical_circuits)" title="Network analysis (electrical circuits)">electrical network analysis</a>, and most especially in connection with <a href="/wiki/Filter_(signal_processing)" title="Filter (signal processing)">filter circuits</a> in the transition between a <a href="/wiki/Passband" title="Passband">passband</a> and a <a href="/wiki/Stopband" title="Stopband">stopband</a>. It is most typically applied to the <a href="/wiki/Insertion_loss" title="Insertion loss">insertion loss</a> of the network, but can, in principle, be applied to any relevant function of frequency, and any technology, not just electronics. It is usual to measure roll-off as a function of <a href="/wiki/Logarithmic_scale" title="Logarithmic scale">logarithmic</a> frequency; consequently, the units of roll-off are either <a href="/wiki/Decibel" title="Decibel">decibels</a> per <a href="/wiki/Decade_(log_scale)" title="Decade (log scale)">decade</a> (dB/decade), where a decade is a tenfold increase in frequency, or decibels per <a href="/wiki/Octave_(electronics)" title="Octave (electronics)">octave</a> (dB/8ve), where an octave is a twofold increase in frequency. </p><p>The concept of roll-off stems from the fact that in many networks roll-off tends towards a constant gradient at frequencies well away from the <a href="/wiki/Cut-off_frequency" class="mw-redirect" title="Cut-off frequency">cut-off</a> point of the frequency curve. Roll-off enables the cut-off performance of such a filter network to be reduced to a single number. Note that roll-off can occur with decreasing frequency as well as increasing frequency, depending on the <a href="/wiki/File:Bandform_template.svg" title="File:Bandform template.svg">bandform</a> of the filter being considered: for instance a <a href="/wiki/Low-pass_filter" title="Low-pass filter">low-pass filter</a> will roll-off with increasing frequency, but a <a href="/wiki/High-pass_filter" title="High-pass filter">high-pass filter</a> or the lower <a href="/wiki/Stopband" title="Stopband">stopband</a> of a <a href="/wiki/Band-pass_filter" title="Band-pass filter">band-pass filter</a> will roll-off with decreasing frequency. For brevity, this article describes only low-pass filters. This is to be taken in the spirit of <a href="/wiki/Prototype_filter" title="Prototype filter">prototype filters</a>; the same principles may be applied to high-pass filters by interchanging phrases such as "above cut-off frequency" and "below cut-off frequency". </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="First-order_roll-off">First-order roll-off</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Roll-off&action=edit&section=1" title="Edit section: First-order roll-off"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:First_order_RC_circuit.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/First_order_RC_circuit.svg/250px-First_order_RC_circuit.svg.png" decoding="async" width="250" height="181" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/First_order_RC_circuit.svg/375px-First_order_RC_circuit.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/42/First_order_RC_circuit.svg/500px-First_order_RC_circuit.svg.png 2x" data-file-width="426" data-file-height="308" /></a><figcaption>First-order RC filter <a href="/wiki/Low-pass_filter" title="Low-pass filter">low-pass filter</a> circuit.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Roll-off_graph_6dB.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Roll-off_graph_6dB.svg/250px-Roll-off_graph_6dB.svg.png" decoding="async" width="250" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Roll-off_graph_6dB.svg/375px-Roll-off_graph_6dB.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/88/Roll-off_graph_6dB.svg/500px-Roll-off_graph_6dB.svg.png 2x" data-file-width="1116" data-file-height="753" /></a><figcaption>Roll-off of a first-order low-pass filter is 20 dB/decade (≈6 dB/octave)</figcaption></figure> <p>A simple <a href="/wiki/First-order_linear_differential_equation" class="mw-redirect" title="First-order linear differential equation">first-order</a> network such as a <a href="/wiki/RC_circuit" title="RC circuit">RC circuit</a> will have a roll-off of 20 dB/decade. This is a little over 6 dB/octave and is the more usual description given for this roll-off. This can be shown to be so by considering the voltage <a href="/wiki/Transfer_function" title="Transfer function">transfer function</a>, <i>A</i>, of the RC network:<sup id="cite_ref-Jacob_1-0" class="reference"><a href="#cite_note-Jacob-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\frac {V_{o}}{V_{i}}}={\frac {1}{1+i\omega RC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mi>ω</mi> <mi>R</mi> <mi>C</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {V_{o}}{V_{i}}}={\frac {1}{1+i\omega RC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adda4997951f0bd704b1e54fad86b5a46558d32b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.779ex; height:5.676ex;" alt="{\displaystyle A={\frac {V_{o}}{V_{i}}}={\frac {1}{1+i\omega RC}}}"></span></dd></dl> <p><a href="/wiki/Prototype_filter#Frequency_scaling" title="Prototype filter">Frequency scaling</a> this to <i>ω</i><sub>c</sub> = 1/<i>RC</i> = 1 and forming the power ratio gives, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |A|^{2}={\frac {1}{1+\left({\omega \over \omega _{c}}\right)^{2}}}={\frac {1}{1+\omega ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |A|^{2}={\frac {1}{1+\left({\omega \over \omega _{c}}\right)^{2}}}={\frac {1}{1+\omega ^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eedbc0e891e028509cf4eb34a9943d53a115bdb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:28.896ex; height:8.343ex;" alt="{\displaystyle |A|^{2}={\frac {1}{1+\left({\omega \over \omega _{c}}\right)^{2}}}={\frac {1}{1+\omega ^{2}}}}"></span></dd></dl> <p>In decibels this becomes, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10\log \left({\frac {1}{1+\omega ^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10\log \left({\frac {1}{1+\omega ^{2}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60dd9467cf7fd87586b2c7e73bef21762415f22f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.444ex; height:6.176ex;" alt="{\displaystyle 10\log \left({\frac {1}{1+\omega ^{2}}}\right)}"></span></dd></dl> <p>or expressed as a loss, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=10\log \left({1+\omega ^{2}}\right)\ \mathrm {dB} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mn>10</mn> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=10\log \left({1+\omega ^{2}}\right)\ \mathrm {dB} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c801157d5895ca740b00b52503346e9b71fbeb6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.517ex; height:3.343ex;" alt="{\displaystyle L=10\log \left({1+\omega ^{2}}\right)\ \mathrm {dB} }"></span></dd></dl> <p>At frequencies well above <i>ω</i>=1, this simplifies to, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L\approx 10\log \left(\omega ^{2}\right)=20\log \omega \ \mathrm {dB} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>≈</mo> <mn>10</mn> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mn>20</mn> <mi>log</mi> <mo>⁡</mo> <mi>ω</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L\approx 10\log \left(\omega ^{2}\right)=20\log \omega \ \mathrm {dB} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b1cda5454a1ebaf09809b7e1e6ea710e08c00bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.129ex; height:3.343ex;" alt="{\displaystyle L\approx 10\log \left(\omega ^{2}\right)=20\log \omega \ \mathrm {dB} }"></span></dd></dl> <p>Roll-off is given by, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta L=20\log \left({\omega _{2} \over \omega _{1}}\right)\ \mathrm {dB/interval_{2,1}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>L</mi> <mo>=</mo> <mn>20</mn> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">a</mi> <msub> <mi mathvariant="normal">l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta L=20\log \left({\omega _{2} \over \omega _{1}}\right)\ \mathrm {dB/interval_{2,1}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dbf167f77fd77136e135f94aedc1a801623597a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.898ex; height:6.176ex;" alt="{\displaystyle \Delta L=20\log \left({\omega _{2} \over \omega _{1}}\right)\ \mathrm {dB/interval_{2,1}} }"></span></dd></dl> <p>For a decade this is; </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta L=20\log 10=20\ \mathrm {dB/decade} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>L</mi> <mo>=</mo> <mn>20</mn> <mi>log</mi> <mo>⁡</mo> <mn>10</mn> <mo>=</mo> <mn>20</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta L=20\log 10=20\ \mathrm {dB/decade} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a08353cc73ae7046d9a4531d94d9c50bd95045e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.962ex; height:2.843ex;" alt="{\displaystyle \Delta L=20\log 10=20\ \mathrm {dB/decade} }"></span></dd></dl> <p>and for an octave, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta L=20\log 2\approx 6.0206\ \mathrm {dB/8ve} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>L</mi> <mo>=</mo> <mn>20</mn> <mi>log</mi> <mo>⁡</mo> <mn>2</mn> <mo>≈</mo> <mn>6.0206</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>8</mn> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">e</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta L=20\log 2\approx 6.0206\ \mathrm {dB/8ve} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/589e16f7fb49978a7fd5101aedaa92dd32db7fc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.512ex; height:2.843ex;" alt="{\displaystyle \Delta L=20\log 2\approx 6.0206\ \mathrm {dB/8ve} }"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Higher_order_networks">Higher order networks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Roll-off&action=edit&section=2" title="Edit section: Higher order networks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:High_order_buffered_RC_circuit.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/High_order_buffered_RC_circuit.svg/500px-High_order_buffered_RC_circuit.svg.png" decoding="async" width="500" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/High_order_buffered_RC_circuit.svg/750px-High_order_buffered_RC_circuit.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/High_order_buffered_RC_circuit.svg/1000px-High_order_buffered_RC_circuit.svg.png 2x" data-file-width="1131" data-file-height="308" /></a><figcaption>Multiple order RC filter buffered between stages.</figcaption></figure> <figure class="mw-halign-none" typeof="mw:File/Thumb"><a href="/wiki/File:Roll-off_graph_multiple.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Roll-off_graph_multiple.svg/250px-Roll-off_graph_multiple.svg.png" decoding="async" width="250" height="145" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Roll-off_graph_multiple.svg/375px-Roll-off_graph_multiple.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/94/Roll-off_graph_multiple.svg/500px-Roll-off_graph_multiple.svg.png 2x" data-file-width="1299" data-file-height="753" /></a><figcaption>Roll-off graph of higher-order low-pass filters showing various rates of roll-off</figcaption></figure> <p>A higher order network can be constructed by cascading first-order sections together. If a <a href="/wiki/Unity_gain_buffer_amplifier" class="mw-redirect" title="Unity gain buffer amplifier">unity gain buffer amplifier</a> is placed between each section (or some other <a href="/wiki/Active_filter" title="Active filter">active topology</a> is used) there is no interaction between the stages. In that circumstance, for <i>n</i> identical first-order sections in cascade, the voltage transfer function of the complete network is given by;<sup id="cite_ref-Jacob_1-1" class="reference"><a href="#cite_note-Jacob-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\mathrm {T} }=A^{n}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msub> <mo>=</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\mathrm {T} }=A^{n}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0366aab6a21c3c06671f610e1ac9530017d02b97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.803ex; height:2.676ex;" alt="{\displaystyle A_{\mathrm {T} }=A^{n}\ }"></span></dd></dl> <p>consequently, the total roll-off is given by, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta L_{\text{T}}=n\,\Delta L=20n{\text{ dB/decade}}\approx 6n{\text{ dB/octave}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> <mo>=</mo> <mi>n</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mi>L</mi> <mo>=</mo> <mn>20</mn> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> dB/decade</mtext> </mrow> <mo>≈</mo> <mn>6</mn> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> dB/octave</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta L_{\text{T}}=n\,\Delta L=20n{\text{ dB/decade}}\approx 6n{\text{ dB/octave}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b22ff78b5657f6ce2be4ceefe74d81e61a2c6608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.539ex; height:2.843ex;" alt="{\displaystyle \Delta L_{\text{T}}=n\,\Delta L=20n{\text{ dB/decade}}\approx 6n{\text{ dB/octave}}}"></span></dd></dl> <p>A similar effect can be achieved in the <a href="/wiki/Digital_filter" title="Digital filter">digital domain</a> by repeatedly applying the same filtering algorithm to the signal.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:LC_ladder_circuit.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/LC_ladder_circuit.svg/500px-LC_ladder_circuit.svg.png" decoding="async" width="500" height="148" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/LC_ladder_circuit.svg/750px-LC_ladder_circuit.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/LC_ladder_circuit.svg/1000px-LC_ladder_circuit.svg.png 2x" data-file-width="1176" data-file-height="349" /></a><figcaption>LC low-pass ladder circuit. Each element (that is L or C) adds an order to the filter and a <a href="/wiki/Pole_(complex_analysis)" class="mw-redirect" title="Pole (complex analysis)">pole</a> to the <a href="/wiki/Driving_point_impedance" class="mw-redirect" title="Driving point impedance">driving point impedance</a>.</figcaption></figure> <p>The calculation of transfer function becomes somewhat more complicated when the sections are not all identical, or when the popular <a href="/wiki/Ladder_topology" class="mw-redirect" title="Ladder topology">ladder topology</a> construction is used to realise the filter. In a ladder filter each section of the filter has an effect on its immediate neighbours and a lesser effect on more remote sections so the response is not a simple <i>A<sup>n</sup></i> even when all the sections are identical. For some filter classes, such as the <a href="/wiki/Butterworth_filter" title="Butterworth filter">Butterworth filter</a>, the insertion loss is still <a href="/wiki/Monotonic_function" title="Monotonic function">monotonically</a> increasing with frequency and quickly <a href="/wiki/Asymptote" title="Asymptote">asymptotically</a> converges to a roll-off of 20<i>n</i> dB/decade, but in others, such as the <a href="/wiki/Chebyshev_filter" title="Chebyshev filter">Chebyshev</a> or <a href="/wiki/Elliptic_filter" title="Elliptic filter">elliptic filter</a> the roll-off near the cut-off frequency is much faster and elsewhere the response is anything but monotonic. Nevertheless, all filter classes eventually converge to a roll-off of 20<i>n</i> dB/decade theoretically at some arbitrarily high frequency, but in many applications this will occur in a frequency band of no interest to the application and <a href="/wiki/Parasitic_capacitance" title="Parasitic capacitance">parasitic effects</a> may well start to dominate long before this happens.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Roll-off&action=edit&section=3" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Filters with a high roll-off were first developed to prevent crosstalk between adjacent channels on telephone <a href="/wiki/Frequency_division_multiplexing" class="mw-redirect" title="Frequency division multiplexing">FDM</a> systems.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Roll-off is also significant on audio loudspeaker <a href="/wiki/Audio_crossover" title="Audio crossover">crossover filters</a>: here the need is not so much for a high roll-off but that the roll-offs of the high frequency and low-frequency sections are symmetrical and complementary. An interesting need for high roll-off arises in <a href="/wiki/EEG" class="mw-redirect" title="EEG">EEG</a> machines. Here the filters mostly make do with a basic 20 dB/decade roll-off, however, some instruments provide a switchable 35 Hz filter at the high frequency end with a faster roll-off to help filter out noise generated by muscle activity.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Roll-off&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Bode_plot" title="Bode plot">Bode plot</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Roll-off&action=edit&section=5" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Jacob-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Jacob_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Jacob_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">J. Michael Jacob, <i>Advanced AC circuits and electronics: principles & applications</i>, pages 150-152, Cengage Learning 2003 <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7668-2330-X" title="Special:BookSources/0-7668-2330-X">0-7668-2330-X</a>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Todd, pp 107–108</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Giovanni Bianchi, Roberto Sorrentino, <i>Electronic filter simulation & design</i>, pages 129–130, McGraw-Hill Professional 2007 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-149467-7" title="Special:BookSources/0-07-149467-7">0-07-149467-7</a>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Lundheim, L, "On Shannon and "Shannon's Formula", <i>Telektronikk</i>, <b>vol. 98</b>, no. 1, 2002, pp. 24–25.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Mayer et al, pp 104–105.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Roll-off&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>J. William Helton, Orlando Merino, <i>Classical control using H [infinity] methods: an introduction to design</i>, pages 23–25, Society for Industrial and Applied Mathematics 1998 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-89871-424-9" title="Special:BookSources/0-89871-424-9">0-89871-424-9</a>.</li> <li>Todd C. Handy, <i>Event-related potentials: a methods handbook</i>, pages 89–92, 107–109, MIT Press 2004 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-262-08333-7" title="Special:BookSources/0-262-08333-7">0-262-08333-7</a>.</li> <li>Fay S. Tyner, John Russell Knott, W. Brem Mayer (ed.), <i>Fundamentals of EEG Technology: Basic concepts and methods</i>, pages 101–102, Lippincott Williams & Wilkins 1983 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-89004-385-X" title="Special:BookSources/0-89004-385-X">0-89004-385-X</a>.</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Roll-off&oldid=1254463619">https://en.wikipedia.org/w/index.php?title=Roll-off&oldid=1254463619</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Electronic_design" title="Category:Electronic design">Electronic design</a></li><li><a href="/wiki/Category:Tone,_EQ_and_filter" title="Category:Tone, EQ and filter">Tone, EQ and filter</a></li><li><a href="/wiki/Category:Filter_frequency_response" title="Category:Filter frequency response">Filter frequency response</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 31 October 2024, at 03:07<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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