In the opamp circuit shown, the Zener diodes Z1 and Z2 clamp the output voltage V_o to +5 V or -5 V. The switch S is intially closed and is opened at time t = 0

The time t = t_{1 (in seconds) at which Vo changes state is _____.}

An opamp has a finite open loop voltage gain of 100. Its input offset voltage V_ios (= +5mV) is modeled as shown in the circuit below. The amplifier is ideal in all other respects. V_input is 25 mV.

The output voltage (in millivolts) is ________.

A speech signal is sampled at 8 kHz and encoded into PCM format using 8 bits/sample. The PCM data is transmitted through a baseband channel via 4-level PAM. The minimum bandwidth (in kHz) required for transmission is _________.

An information source generates a binary sequence $$\left\{ {{\alpha _n}} \right\}.{\alpha _n}$$ can take one of the two possible values −1 and +1 with equal probability and are statistically independent and identically distributed. This sequence is pre-coded to obtain another sequence $$\left\{ {{\beta _n}} \right\},$$ as $${\beta _n} = {\alpha _n} + k{\mkern 1mu} {\alpha _{n - 3}}$$ . The sequence $$\left\{ {{\beta _n}} \right\}$$ is used to modulate a pulse $$g(t)$$ to generate the baseband signal

$$x\left( t \right) = \sum\limits_{n = - \infty }^\infty {{\beta _n}g\left( {t - nT} \right),} $$ where $$g\left( t \right) = \left\{ {\matrix{ {1,} & {0 \le t \le T} \cr 0 & {otherwise} \cr } } \right.$$

If there is a null at $$f = {1 \over {3T}}$$ in the power spectral density of $$X(t)$$, then $$k$$ is _________.