# Sine and Impulse Response Sampled 1st Order Type 0

Cuthbert Nyack
Applet shows the sinusoidal and impulse response for a first order, type 0 system. The difference equation for the sinusoidal response is shown below. eg parameters (5.0, 2.0, 0.4, 100.0, 0.1). At this low frequency, the phase lag is ~0° but the output is smaller than the input by K/(K + 1) (ie the offset).
eg parameters (5.0, 2.0, 0.03, 100.0, 3.0). The applied frequency is equal to the 3dB frequency (1 + K)/T1 = 3.0. The phase lag is 45° and the attenuation is 0.7.
eg parameters (5.0, 0.7, 0.13, 100.0, 0.9). The pole is at zero and the output is a sinusoid from the start. No transient buildup.
eg parameters (4.9, 1.0, 0.42, 100.0, 0.2). The pole is just outside the unit circle and the output shows a growing oscillation at half the sampling frequency.

The impulse response is given by the following equation. eg parameters (5.0, 5.0, 0.1, 50.0, 1.0). The pole is close to +1 and the sampled response follows the continuous response.
eg parameters (5.0, 1.0, 0.18, 50.0). The pole is close to 0 and the sampled response is the unit impulse.
eg parameters (3.0, 0.4, 0.27, 50.0, 0.25). The pole is close to -1 and the sampled response is a decaying oscillation at half the sampling frequency.
eg parameters (3.0, 0.4, 0.29, 50.0, 0.02). The pole is just outside the unit circle and the output is a growing oscillation at half the sampling frequency.

Gif images below show how the applets should appear.  