# Sine and Impulse Response Sampled 1st Order Type 1

Cuthbert Nyack
Applet shows the sinusoidal and impulse response for a first order, type 1 system. The difference equation for the sinusoidal response is shown below. Eg parameters (5.0, 2.0, 0.63, 100.0, 0.1). At this low frequency the output is almost equal to the input and the phase lag is small.
Eg parameters (5.0, 2.0, 0.03, 100.0, 2.5). Here the input frequency is the 3db frequency (K/T1 = 2.5), the output lags the input by 45° and the amplitude is down by ~0.7.
Eg parameters (4.0, 2.0, 0.03, 100.0, 4.0). Here the input frequency is twice the 3db frequency, the amplitude is down by 0.45 and the phase lag is ~63° .

Eg parameters (4.0, 1.0, 0.5, 100.0, 0.3). Here the pole is at -1 and the output shows oscillations at half the sampling frequency.
Eg parameters (4.0, 1.0, 0.51, 100.0, 0.3). Here the pole is outside the unit circle and the system is unstable.

Eg parameters (2.0, 1.0, 1.0, 30.0, 3.14). Here the pole is at -1 and the output should show oscillations at half the sampling frequency. However since the applied frequency is also at half the sampling frequency, then the result is a linearly increasing amplitude at half the sampling frequency.

The impulse response is given by the following equation. eg parameters (1.0, 1.0, 0.1, 50.0). The pole is close to +1 and the impulse response is similar to the continuous case.
eg parameters (1.0, 1.0, 1.0, 50.0). The pole is at the origin and the impulse response is zero after the first output.
eg parameters (1.0, 1.0, 2.0, 50.0). The pole is at -1 and the impulse response is an oscillation at half the sampling frequency.
eg parameters (1.1, 1.0, 1.83, 50.0). The pole is just outside the unit circle and the response is unstable.

Gif images below show how the applets should appear.  