In this paper we present our solid-state nanosecond pulse source (the solid-state impedance-matched Marx generator) which can generate arbitrary waveforms and which can be used for pulsed discharge generation. The purpose of the development of such a generator is twofold: by being able to change the waveform at will, we aim to control the discharge generated by such pulses very precisely which can be very useful for plasma applications, but also for more fundamental studies. In the presented study, we applied the arbitrary-waveform pulse source for streamer discharge generation in a cylinder-wire-like arrangement and used the arbitrary-waveform capability to change the rise time (in our experiments we used 6.8-26.2 ns) of unipolar positive pulses with 6-10 kV amplitude and 80 ns duration. Additionally, we introduced variations of a step in the rising edge of the waveform. We performed measurements both in air and nitrogen to electrically characterize the discharge while analyzing the streamer propagation in the plasma reactor with intensified charge-coupled device imaging and measured ozone generation (in air). The results show that we can indeed control the propagation of the streamer discharge with the stepped waveform, but that the rise-time variation has little effect on the streamer propagation in our system. However, the streamer velocity and structure differs significantly comparing discharges in nitrogen and air for the same applied voltage waveform. Additionally, for some of the stepped waveforms we found a slight increase of the ozone yield for air at low overall energy densities.