Free-electron lasers (FELs) are among the world's most intense artificial artificial sources of coherent light and are tunable to various wavelengths, including the X-ray spectrum. X-ray FELs (XFELs) are extremely useful for diffraction experiments to study molecules, materials, and quantum systems. A FEL consists of an electron accelerator and a structure of magnets called an undulator. The undulator has a periodic magnetic field, and when an electron beam passes through the undulator, the Lorentz force forces the electrons to oscillate and emit what is known asspontaneous undulator radiation. Initially, the undulator radiation is spontaneously emitted and incoherent. However, aAs the electrons interact with this initial spontaneous undulator radiation, they change their relative positions and form micro-bunches of electrons. These microbunches are shorter than the undulator radiation wavelength. Hence, the waves emitted by the electrons from the same microbunch arethey become in phase, meaning the radiation is now coherent with the radiation field, and the state of coherence develops. This process is known as self-amplified spontaneous emission (SASE). Due to the coherence, tThe radiation intensity grows exponentially along the undulator, forming several peaks in the radiation pulse known as SASE spikes. One technique for obtaining ultra-short laser pulses is to isolate single SASE spikes by controlling where, along the electron beam, the SASE spikes can grow. This growth limitation is archieved by modulating the electron energies, thus only allowing electrons at specific positions along the electron beam to radiate. In addition, to keep positive interference between undulator radiation from electrons with different energies, the energy modulation must be compensated with a gradient of the magnetic field amplitude of the undulator, so-called tapering. There are plans to implement this technique at one of the beamlines at the European X-ray FEL (EuXFEL) to generate attosecond X-ray pulses and study quantum systems. One goal of the design process is to choose design parameters for the electron beam's modulation amplitude and the undulator's tapering coefficient. These design parameters shall be chosen so that the XFEL will have as short pulse duration as possible while at the same time not getting too low peak power. This thesis aims to study the effect of electron energy modulation and undulator tapering on the SASE and how the modulation amplitude and the tapering coefficient affect the XFEL's peak power and pulse duration. A model was developed to simulate SASE with a modulated electron beam in a tapered undulator. With this model, a parameter scan gave the average peak power and pulse duration as functions of the modulation amplitude and the tapering coefficient. The parameter scan showed that the peak power and the pulse duration decrease as the modulation amplitude and the tapering coefficient increase. Therefore, a trade-off exists between high peak power and short pulse duration. It was possible to exclude sets of the parameters that gave too low peak power or long pulse duration. This study also found an optimum range for the tapering coefficient where the peak power had a local maximum without a significant increase in pulse duration. The physics behind this optimal tapering coefficient is also discussed in connection to the electrons' energy modulation.