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style='min-width: 970px; min-height: 90px;'><script>googletag.cmd.push(function() { googletag.display("jumbo-banner-ad-container"); });</script></div></div> <div class="site-inner"><div class="content-sidebar-wrap"><main class="content" id="genesis-content"><div class="breadcrumb">You are here: <span class="breadcrumb-link-wrap"><a class="breadcrumb-link" href="https://www.testandmeasurementtips.com/"><span class="breadcrumb-link-text-wrap">Home</span></a><meta ></span> <span aria-label="breadcrumb separator">/</span> <span class="breadcrumb-link-wrap"><a class="breadcrumb-link" href="https://www.testandmeasurementtips.com/category/faq/"><span class="breadcrumb-link-text-wrap">FAQ</span></a><meta ></span> <span aria-label="breadcrumb separator">/</span> How to determine noise figure: part 2</div><article class="post-19265 post type-post status-publish format-standard has-post-thumbnail category-faq category-featured category-power-sensing category-vector-network-analyzer category-vector-network-analyzers tag-faq entry" aria-label="How to determine noise figure: part 2"><header class="entry-header"><h1 class="entry-title">How to determine noise figure: part 2</h1> <p class="entry-meta"><time class="entry-time">November 13, 2024</time> By <span class="entry-author"><a href="https://www.testandmeasurementtips.com/author/rick-nelson/" class="entry-author-link" rel="author"><span class="entry-author-name">Rick Nelson</span></a></span> <span class="entry-comments-link"><a href="https://www.testandmeasurementtips.com/how-to-determine-noise-figure-part-2/#respond">Leave a Comment</a></span> </p></header><div class="entry-content"><div class="addtoany_share_save_container addtoany_content addtoany_content_top"><div class="a2a_kit a2a_kit_size_16 addtoany_list" data-a2a-url="https://www.testandmeasurementtips.com/how-to-determine-noise-figure-part-2/" data-a2a-title="How to determine noise figure: part 2"><a class="a2a_button_facebook" href="https://www.addtoany.com/add_to/facebook?linkurl=https%3A%2F%2Fwww.testandmeasurementtips.com%2Fhow-to-determine-noise-figure-part-2%2F&linkname=How%20to%20determine%20noise%20figure%3A%20part%202" title="Facebook" rel="nofollow noopener" target="_blank"></a><a class="a2a_button_x" href="https://www.addtoany.com/add_to/x?linkurl=https%3A%2F%2Fwww.testandmeasurementtips.com%2Fhow-to-determine-noise-figure-part-2%2F&linkname=How%20to%20determine%20noise%20figure%3A%20part%202" title="X" rel="nofollow noopener" target="_blank"></a><a class="a2a_button_linkedin" href="https://www.addtoany.com/add_to/linkedin?linkurl=https%3A%2F%2Fwww.testandmeasurementtips.com%2Fhow-to-determine-noise-figure-part-2%2F&linkname=How%20to%20determine%20noise%20figure%3A%20part%202" title="LinkedIn" rel="nofollow noopener" target="_blank"></a><a class="a2a_dd addtoany_share_save addtoany_share" href="https://www.addtoany.com/share"></a></div></div><p><em>The relationship between noise and temperature prompted a precursor of the IEEE to promulgate an alternative definition of noise figure in 1959.</em></p> <figure id="attachment_19266" aria-describedby="caption-attachment-19266" style="width: 184px" class="wp-caption aligncenter"><a href="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-11.50.34-AM.png"><img decoding="async" class="wp-image-19266" src="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-11.50.34-AM.png" alt="" width="184" height="97" /></a><figcaption id="caption-attachment-19266" class="wp-caption-text">Equation 1.</figcaption></figure> <p>In <a href="https://www.testandmeasurementtips.com/how-to-determine-noise-figure-part-1/" target="_blank" rel="noopener">part 1</a> of this series, we described the work of the Danish-American radio engineer Harald Friis, who described noise factor <em>F</em> of a device or system as the ratio of the input-power signal-to-noise ratio (SNR) to the output-power SNR. If <em>S<sub>IN</sub></em>, <em>S<sub>OUT</sub></em>, <em>N<sub>IN</sub></em>, and <em>N<sub>OUT</sub></em> are the input and output signal levels and the input and output noise levels, respectively, then also note that noise <em>figure</em> (<em>NF</em>) is generally expressed in decibels:</p> <figure id="attachment_19267" aria-describedby="caption-attachment-19267" style="width: 172px" class="wp-caption aligncenter"><a href="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-11.51.19-AM.png"><img decoding="async" class="wp-image-19267" src="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-11.51.19-AM.png" alt="noise" width="172" height="40" /></a><figcaption id="caption-attachment-19267" class="wp-caption-text">Equation 2.</figcaption></figure> <p>Finally, we noted that if <em>S<sub>IN</sub></em>, <em>S<sub>OUT</sub></em>, <em>N<sub>IN</sub></em>, and <em>N<sub>OUT</sub></em> are all decibel values, then <em>NF</em> in dB is</p> <figure id="attachment_19268" aria-describedby="caption-attachment-19268" style="width: 290px" class="wp-caption aligncenter"><a href="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-11.52.29-AM.png"><img loading="lazy" decoding="async" class="wp-image-19268" src="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-11.52.29-AM.png" alt="noise" width="290" height="39" srcset="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-11.52.29-AM.png 522w, https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-11.52.29-AM-300x40.png 300w" sizes="auto, (max-width: 290px) 100vw, 290px" /></a><figcaption id="caption-attachment-19268" class="wp-caption-text">Equation 3.</figcaption></figure> <p><strong>You mentioned that we would be looking at another aspect of <em>NF</em>.<br /> </strong>Yes. We can look at noise figures from the perspective of a fundamental noise cause — temperature. In the late 1920s, <a href="https://www.testandmeasurementtips.com/noise-quantification-and-measurement-faq/" target="_blank" rel="noopener">Schottky and Johnson</a> hypothesized and measured the thermal noise phenomenon, in which an open-circuit resistor develops a noise voltage <em>v<sub>N</sub></em> that varies with temperature. Shortly after, <a href="https://www.testandmeasurementtips.com/the-difference-between-signal-under-sampling-aliasing-and-folding-faq/" target="_blank" rel="noopener">Nyquist</a> showed that:</p> <figure id="attachment_19269" aria-describedby="caption-attachment-19269" style="width: 173px" class="wp-caption aligncenter"><a href="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-11.53.07-AM.png"><img loading="lazy" decoding="async" class="wp-image-19269" src="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-11.53.07-AM.png" alt="" width="173" height="66" /></a><figcaption id="caption-attachment-19269" class="wp-caption-text">Equation 4.</figcaption></figure> <p>where <em>T</em> equals the temperature in Kelvin, <em>R</em> equals the ohms resistance, B equals hertz’s measurement bandwidth, and k equals <a href="https://www.testandmeasurementtips.com/measuring-entropy-faq/" target="_blank" rel="noopener">Boltzmann’s constant</a> (1.38 ´ 10<sup>-23</sup> Joules/Kelvin). Essentially, a resistor becomes a voltage source of level <em>v<sub>N</sub></em> in series with a source resistance with a value of <em>R<sub>S</sub></em>, as shown in <strong>Figure 1a</strong>.</p> <figure id="attachment_19274" aria-describedby="caption-attachment-19274" style="width: 1151px" class="wp-caption aligncenter"><a href="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Nelson_NoiseFigure_pt2_fig1.jpg"><img loading="lazy" decoding="async" class="wp-image-19274 size-full" src="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Nelson_NoiseFigure_pt2_fig1.jpg" alt="noise" width="1151" height="695" srcset="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Nelson_NoiseFigure_pt2_fig1.jpg 1151w, https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Nelson_NoiseFigure_pt2_fig1-300x181.jpg 300w, https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Nelson_NoiseFigure_pt2_fig1-1024x618.jpg 1024w, https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Nelson_NoiseFigure_pt2_fig1-768x464.jpg 768w" sizes="auto, (max-width: 1151px) 100vw, 1151px" /></a><figcaption id="caption-attachment-19274" class="wp-caption-text">Figure 1. A resistor modeled as a voltage vN and series resistance RS (a) delivers maximum noise power PN to load RL when RS equals RL (b).</figcaption></figure> <p>In RF/microwave circuits, we are interested in the noise power level <em>P<sub>N</sub></em> that a noise voltage source can deliver to a load—the input to a device under test (DUT), for example (<strong>Figure 1b</strong>). The worst case will occur when the DUT input impedance RL matches RS in accordance with the <a href="https://www.testandmeasurementtips.com/thevenin-norton-help-interpret-measurement-results/">maximum power transfer theorem</a>. If we set both equal to <em>R</em>, we can calculate the noise current:</p> <p><a href="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-11.54.16-AM.png"><img loading="lazy" decoding="async" class="aligncenter wp-image-19270" src="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-11.54.16-AM.png" alt="" width="120" height="67" /></a></p> <p>Consequently, the maximum noise power transfer into the DUT in watts is</p> <p><a href="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-11.54.56-AM.png"><img loading="lazy" decoding="async" class="aligncenter wp-image-19271" src="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-11.54.56-AM.png" alt="" width="199" height="67" srcset="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-11.54.56-AM.png 326w, https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-11.54.56-AM-300x101.png 300w" sizes="auto, (max-width: 199px) 100vw, 199px" /></a></p> <p>Substituting <strong>Equation 4</strong> for <em>v<sub>N</sub></em> gives us:</p> <figure id="attachment_19272" aria-describedby="caption-attachment-19272" style="width: 205px" class="wp-caption aligncenter"><a href="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-11.57.15-AM.png"><img loading="lazy" decoding="async" class="wp-image-19272" src="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-11.57.15-AM.png" alt="noise" width="205" height="68" srcset="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-11.57.15-AM.png 344w, https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-11.57.15-AM-300x99.png 300w" sizes="auto, (max-width: 205px) 100vw, 205px" /></a><figcaption id="caption-attachment-19272" class="wp-caption-text">Equation 5.</figcaption></figure> <p>The resistances <em>R</em> cancel out, and a bandwidth of 1 Hz is typically used for comparison purposes. Hence, the noise power spectral density PN per hertz depends only on T. A common reference temperature level is 290K (17°C or about 62°F). You can calculate that <em>P<sub>N</sub></em> equals -204 dBW/Hz or -174 dBm/Hz at that temperature.</p> <p><strong>So, how does this apply to the noise figure?<br /> </strong>First, we can rewrite <strong>Equation 1</strong> with <em>G</em> representing the DUT gain (in linear terms, not decibels) and <em>N<sub>ADD</sub></em> equaling the noise power added by the DUT (again in linear terms, not decibels):</p> <figure id="attachment_19273" aria-describedby="caption-attachment-19273" style="width: 296px" class="wp-caption aligncenter"><a href="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-11.59.48-AM.png"><img loading="lazy" decoding="async" class="wp-image-19273" src="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-11.59.48-AM.png" alt="noise" width="296" height="68" srcset="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-11.59.48-AM.png 590w, https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-11.59.48-AM-300x69.png 300w" sizes="auto, (max-width: 296px) 100vw, 296px" /></a><figcaption id="caption-attachment-19273" class="wp-caption-text">Equation 6.</figcaption></figure> <p>Now, we note that the input noise <em>N<sub>IN</sub></em> equals <em>P<sub>N</sub></em> = k<em>TB</em> from <strong>Equation 5</strong>, so we can rewrite <strong>Equation 6</strong> as follows:</p> <figure id="attachment_19276" aria-describedby="caption-attachment-19276" style="width: 281px" class="wp-caption aligncenter"><a href="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-12.01.52-PM.png"><img loading="lazy" decoding="async" class=" wp-image-19276" src="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-12.01.52-PM.png" alt="" width="281" height="60" srcset="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-12.01.52-PM.png 506w, https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-12.01.52-PM-300x64.png 300w" sizes="auto, (max-width: 281px) 100vw, 281px" /></a><figcaption id="caption-attachment-19276" class="wp-caption-text">Equation 7.</figcaption></figure> <p>Then, as was the case for the method described in part 1, we can calculate the noise figure in dB:</p> <figure id="attachment_19277" aria-describedby="caption-attachment-19277" style="width: 216px" class="wp-caption aligncenter"><a href="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-12.02.19-PM.png"><img loading="lazy" decoding="async" class=" wp-image-19277" src="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-12.02.19-PM.png" alt="" width="216" height="43" srcset="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-12.02.19-PM.png 372w, https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-12.02.19-PM-300x60.png 300w" sizes="auto, (max-width: 216px) 100vw, 216px" /></a><figcaption id="caption-attachment-19277" class="wp-caption-text">Equation 8.</figcaption></figure> <p>This temperature approach to noise figure was promulgated in 1959 by the Institute of Radio Engineers, a precursor to the IEEE.</p> <p><strong>Are there any other approaches to specifying noise performance?<br /> </strong>Yes. We can rewrite <strong>Equation 3</strong> as follows:</p> <figure id="attachment_19278" aria-describedby="caption-attachment-19278" style="width: 127px" class="wp-caption aligncenter"><a href="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-12.02.55-PM.png"><img loading="lazy" decoding="async" class=" wp-image-19278" src="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-12.02.55-PM.png" alt="" width="127" height="80" /></a><figcaption id="caption-attachment-19278" class="wp-caption-text">Equation 9.</figcaption></figure> <p>Consider <strong>Figure 2</strong>, where a noise source is connected to an impedance-matched <a href="https://www.testandmeasurementtips.com/key-considerations-for-rf-power-measurement-equipment-faq/" target="_blank" rel="noopener">power sensor and meter</a>. The latter, which we assume contributes minimal noise, tells us the noise source adds noise power <em>N<sub>ADD</sub></em>.</p> <figure id="attachment_19275" aria-describedby="caption-attachment-19275" style="width: 1341px" class="wp-caption aligncenter"><a href="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Nelson_NoiseFigure_pt2_fig2.jpg"><img loading="lazy" decoding="async" class="wp-image-19275 size-full" src="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Nelson_NoiseFigure_pt2_fig2.jpg" alt="noise" width="1341" height="389" srcset="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Nelson_NoiseFigure_pt2_fig2.jpg 1341w, https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Nelson_NoiseFigure_pt2_fig2-300x87.jpg 300w, https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Nelson_NoiseFigure_pt2_fig2-1024x297.jpg 1024w, https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Nelson_NoiseFigure_pt2_fig2-768x223.jpg 768w" sizes="auto, (max-width: 1341px) 100vw, 1341px" /></a><figcaption id="caption-attachment-19275" class="wp-caption-text">Figure 2. A noise source delivers a noise power of NADD to a power meter.</figcaption></figure> <p>We can now answer a key question: If we want to replace our noise source with a resistor, as in Figure 1a, at what temperature — call it <em>T<sub>E</sub></em>, for equivalent noise temperature (sometimes called effective noise temperature, or just noise temperature) — would we need to maintain the resistor for it to generate the same noise power <em>N<sub>ADD</sub></em> that our noise source is generating?</p> <p><strong>Why would we want to do that?<br /> </strong>We wouldn’t want to do that physically, but <em>T<sub>E</sub></em> gives us another figure of merit for specifying the noise performance of a component. We can calculate <em>T<sub>E</sub></em> by substituting <em>N<sub>ADD</sub></em> for <em>P<sub>N</sub></em> in <strong>Equation 9</strong>:</p> <figure id="attachment_19279" aria-describedby="caption-attachment-19279" style="width: 121px" class="wp-caption aligncenter"><a href="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-12.03.35-PM.png"><img loading="lazy" decoding="async" class="wp-image-19279" src="https://www.testandmeasurementtips.com/wp-content/uploads/2024/11/Screen-Shot-2024-11-13-at-12.03.35-PM.png" alt="noise" width="121" height="80" /></a><figcaption id="caption-attachment-19279" class="wp-caption-text">Equation 10.</figcaption></figure> <p><strong>Wow! We’ve got two equations for noise factor (Equations 1 and 7), three equations for noise figure (Equations 2, 3, and 8), and now we have an equation for <em>T<sub>E</sub></em> (Equation 10). When do we use each, and can we convert between them?<br /> </strong>We’ll sort that out in <a href="https://www.testandmeasurementtips.com/how-to-determine-noise-figure-part-3/" target="_blank" rel="noopener">part 3</a>.</p> <h3><strong>For further reading</strong></h3> <p><a href="https://ieeexplore.ieee.org/document/4065915" target="_blank" rel="noopener">IRE Standards on Methods of Measuring Noise in Linear Twoports, 1959</a> (IEEE<em>Xplore</em>)</p> <h3><strong>Related EE World content</strong></h3> <p><a href="https://www.testandmeasurementtips.com/avoid-errors-low-voltage-measurements/" target="_blank" rel="noopener">How to avoid errors in low-voltage measurements</a><br /> <a href="https://www.testandmeasurementtips.com/measuring-phase-noise-and-jitter-faq/" target="_blank" rel="noopener">Measuring phase noise and jitter</a><br /> <a href="https://www.testandmeasurementtips.com/5g-radios-increase-emphasis-on-compliance-testing/" target="_blank" rel="noopener">5G radios 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