We follow the four steps described in Section 7.1.5. The noise spectrum of R is given by S_v( f ) = 4kT R. Next, modeling the noise of R by a series voltage source V_R, we compute the transfer function from V_R  to  V_{out}:

\frac{V_{out}}{V_R}(s) =\frac{1}{RCs + 1}                                 (7.17)

From the theorem in Section 7.1.1, we have

S_{out}( f ) = S_v( f )\left|\frac{V_{out}}{V_R}(jω)\right|^2                   (7.18)

= 4kT R \frac{1}{4π^2R^2C^2 f ^2 + 1}                       (7.19)

Thus, the white noise spectrum of the resistor is shaped by a low-pass characteristic (Fig. 7.16). To calculate the total noise power at the output, we write

P_{n,out} =\int_{0}^{\infty }{\frac{4kT R}{4π^2R^2C^2 f^2 + 1}d f}                                (7.20)

Note that the integration must be with respect to f rather than ω (why?). Since

\int{\frac{dx}{x^2 + 1}} = \tan^{−1} x                              (7.21)

the integral reduces to

\begin{aligned} P_{n, \text { out }} &=\left.\frac{2 k T}{\pi C} \tan ^{-1} u\right|_{u=0} ^{u=\infty}             (7.22) \\ &=\frac{k T}{C} \end{aligned}                    (7.23)

Note that the unit of k T / C is V^2. We may also consider \sqrt{k T / C} as the total rms noise voltage measured at the output. For example, with a 1-pF capacitor, the total noise voltage is equal to 64.3 \mu V_ {rms } at T=300 \mathrm{~K}.

Equation (7.23) implies that the total noise at the output of the circuit shown in Fig. 7.15 is independent of the value of R. Intuitively, this is because for larger values of R, the associated noise per unit bandwidth increases while the overall bandwidth of the circuit decreases. The fact that kT / C noise can be decreased only by increasing C (if T is fixed) introduces many difficulties in the design of analog circuits (Chapter 13).