Non-periodic signals

These records are finite-support pulses, delayed envelopes, or a single linear chirp. They do not repeat every T = 1/f, except that sweep and exponential families still use f_s as the time base.

1. Sweep signal (linear chirp)

Instantaneous frequency increases linearly from f₁ to f₂ over the whole record. Both sliders are 20 discrete steps, each step 5 % of f_s/2. The implementation requires

0 < f₁ < f₂ < f_s / 2 (14)

With normalized time u = n / (N − 1) (or n/N in the generator) and duration T_rec = N/f_s, the instantaneous frequency and phase are

f(u) = f₁ + (f₂ − f₁) u
φ(u) = 2π [ f₁ u T_rec + ½ (f₂ − f₁) u² T_rec ] (15)

x[n] = A sin(φ(n)) + DC (16)

There is no middle “wave frequency” dial — only Amplitude and Sampling frequency, plus the two percentage sliders (read out in hertz).

Background: Wikipedia: Chirp.

2. Exponential signal

A delayed decaying exponential. SigGer uses the sample period

T = 2 / f_s (17)

(not 1/f_s). The start index is n₀ = int(t₀ / T) with t₀ in seconds. Then

x[n] = 0 for n < n₀
x[n] = A exp( − (n − n₀) T / RC ) + DC for n ≥ n₀ (18)

Slider	Range	Conversion
Initial time instant	100 steps, 0–1000 ms	t₀ = (value in ms) / 1000
Exponential time constant	1000 steps, 10–1 000 000 µs	RC = (value in µs) / 10⁶

RC is the analog time constant of e^−t/RC. Larger RC means slower decay. There is no waveform-frequency dial.

3. Damped sine signal

Same delay and envelope as (18), multiplied by a sinusoid whose frequency f is again Nyquist-limited. The middle dial (Sine frequency) is restored in the periodic position: Amplitude | Sine frequency | Sampling frequency.

x[n] = A e^{−((n−n₀)T)/RC} sin( 2π f (n−n₀) / f_s ) + DC (n ≥ n₀) (19)

For n < n₀, x[n] = DC only (the oscillating part is zero). Sliders match the exponential window (milliseconds and microseconds).

4. Discrete-time exponential

A geometric sequence (the exact discrete counterpart of a sampled exponential when a = e^−T/RC):

x[n] = 0 n < n₀
x[n] = A · a^{(n − n₀)} + DC n ≥ n₀ (20)

Slider	Steps	Range
Initial sample n₀	100	0–1000 (integer index via linear mapping)
Numerical base a	100	−2.0 … +2.0

Negative bases use integer exponents so the sign alternates. By convention 0⁰ = 1. If |a| > 1 the sequence grows and may overflow floating-point or saturate integers. Sampling frequency is kept for the saved file; there is no analog time-constant slider.

See Wikipedia: Geometric progression and Oppenheim & Schafer on discrete exponentials (References).

5. Exponentially damped discrete-time sinusoid

Same sliders as the discrete exponential, plus a middle dial Digital frequency ω ∈ [0, π] rad/sample (labelled rad/s on the faceplate). Sampling frequency remains on the right.

x[n] = A · a^(n−n₀) sin( ω (n − n₀) ) + DC (n ≥ n₀) (21)

ω = π is the discrete Nyquist frequency (alternating sequence …, +1, −1, +1, … when a = 1). ω = 0 yields a non-oscillating geometric sequence (the sine factor is zero for n > n₀).

Background: Normalized (digital) frequency.

6. Impulse signal

A single Kronecker impulse. The location slider has 100 steps from 0 % to 100 % of the record. The sample index is

n₀ = round( (p / 100) · (N − 1) ) (22)

x[n] = A + DC if n = n₀; DC otherwise (23)

The Initial-time slider of the exponential window is removed; the remaining slider sits on the right and is titled Location where the impulse occurs.

7. Step signal

Same window as the impulse (location 0–100 % of N), synthesizing the discrete unit step u[n − n₀]:

x[n] = DC if n < n₀
x[n] = A + DC if n ≥ n₀ (24)

The slider title is Location where the step occurs.

8. Rectangular pulse signal

Slider	Steps	Mapped quantity
Initial sample	100, 0–50 % of N	n₀ = round((p/100)·(N−1))
Pulse length	100, 0–100 % of N	W = round((p/100)·N) samples

x[n] = A + DC if n₀ ≤ n < n₀ + W
x[n] = DC otherwise (25)

If n₀ + W exceeds N, the pulse is clipped at the end of the record. W = 0 yields a constant DC.

9. Triangle pulse signal

Identical controls to the rectangular pulse. The support [n₀, n₀+W) is filled with an isosceles tent of peak A. With u = (n − n₀) / W (and W ≥ 2),

w(u) = 2u if 0 ≤ u < ½; w(u) = 2(1 − u) if ½ ≤ u < 1
x[n] = A w(u) + DC (26)

W = 1 is treated as a single impulse of height A.

10. Trapezoidal pulse signal

Based on the trapezoidal-wave window, with these changes:

Control	Role
Plateau duration (middle dial)	Linear 0–90 % of N (replaces trapezoidal frequency)
Location where the pulse occurs (left slider)	100 steps, 0–50 % of N → n₀
Rise time (center slider)	Same 5 % steps as the wave editor; unit is % of N
Fall time (right slider)	Same; rise + fall < 100 % of N

Let N_r, N_p, N_f be the rise, plateau, and fall lengths in samples. Then, for k = n − n₀,

rise 0 ≤ k < N_r: x = A (k / N_r) + DC
plateau N_r ≤ k < N_r+N_p: x = A + DC
fall next N_f samples: x = A (1 − k_fall/N_f) + DC
else: x = DC (27)

Segments that run past N − 1 are clipped. A zero rise or fall makes that edge instantaneous.