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We investigated the magnetic-field behavior of the off-diagonal impedance in Co-based amorphous wires under sinusoidal (50 MHz) and pulsed (5 ns rise time) current excitations. For comparison, we measured the field characteristics of the diagonal impedance as well. In general, when an alternating current is applied to a magnetic wire, the voltage signal is generated not only across the wire but also in a pickup coil wound on it. These voltages are related to the diagonal and off-diagonal impedances, respectively. We demonstrate that these impedances have a different behavior as functions of axial magnetic field: the diagonal impedance is symmetrical, whereas the off-diagonal one is antisymmetrical with a near-linear portion within a certain field interval. For the off-diagonal response, the dc bias current is necessary to eliminate circular domains. In the case of the sinusoidal excitation without a dc bias current, the off-diagonal response is very small and irregular. In contrast, the pulsed excitation, combining both high- and low-frequency harmonics, produces the off-diagonal voltage response without additional biasing. This behavior is ideal for a practical sensor circuit design. We discuss the principles of operation of a linear magnetic sensor based on a complementary metal-oxide-semiconductor transistor circuit.