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I'd like to have a sinousid wave with some noise. Then use that signal as input for an arduino, visualize that signal in Matlab and use a simple filter (moving average, for example) to remove the noise. To create the sinusoid I'm using a little DSO Nano v3 but I don't know a simple way to add noise (real noise, I don't want to add noise through Matlab).

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    \$\begingroup\$ Real noise can be created with a reverse biased transistor, then use HPF and amplify noise 1,000 times or more to be useful. Digitally, a true random number generator can emulate noise, though it is more of a 'pop-corn' noise than white or pink noise like a transistor would get you. \$\endgroup\$ – Sparky256 Sep 27 '18 at 1:03
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    \$\begingroup\$ audio signal from an off-station AM radio \$\endgroup\$ – jsotola Sep 27 '18 at 1:05
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This circuit will get you noise; the output noise of Q1 is large enough (5 milliVolts RMS or about 30 mV Peak Peak for 6 sigma or about 1 part per million) to cause serious distortion across the base-emitter junction of Q2, unless Q2 has a large Remitter. [ in the first version of this post, I made a GAIN error; the math shows Q1 gain is 200X, but I erred by simply scaling the total integrated noise in the full bandwidth, or 28 microVolts, by 1,000X. Again correct scale factor is 200X]

schematic

simulate this circuit – Schematic created using CircuitLab

Notice the self-biasing of each stage: scaling the total base resistors to be BETA * Rcollector will cause Vcollector_DC to be nearly VDD/2 under all conditions.

This self-biasing, useful as DC as a operating-point servo-loop (regulator) also NULLS out the very low frequency SIGNAL, because low frequency simply looks like a variation in operating-point. Thus low frequency NOISE will be nulled out.

The bandwidth of the noise is probably set by Rc of stage1 and the Cmiller (input capacitance) of Q2. Wit 40X gain of Q2, and assumed Cob (C_collector_base) of 10pF, thus Cmiller is (1 + Av) * 10pF = 410 pF. With Rc = 1Kohm, and Cload = 10pF (Cob Q1) + 410pF from Q2, the time constant Tau is 1K * 420pF = 0.42 microseconds, radian frequency of 1/0.42uS, Hertz frequency of 1/(2 * pi * 0.42uS) ~~ 400,000Hz. This is not the assumed 1MHz bandwidth. Your noise will be about 3dB (0.707) lower than predicted.

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Additional to what @Sparky256 said, you can simply generate true random noise with a reverse biased diode, followed by a high gain OPAMP. The output of the OPAMP can then be coupled onto the sine wave with a capacitor.

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